Entry Value
Name FINITE_FINITE_PREIMAGE
Conclusion !f t. FINITE t /\ (!y. y IN t ==> FINITE {x | f x = y}) ==> FINITE {x | f x IN t}
Constructive Proof No
Axiom 
!t. t \/ ~t 
(\a. a = (\b. (\c. c) = (\c. c))) (\d. (\e. d e) = d)
Classical Lemmas
  • !x s t. s SUBSET x INSERT t <=> s DELETE x SUBSET t
  • !x s. x IN s ==> x INSERT (s DELETE x) = s
  • !t. (t <=> T) \/ (t <=> F)
  • !f s t. FINITE t /\ (!y. y IN t ==> FINITE {x | x IN s /\ f x = y}) ==> FINITE {x | x IN s /\ f x IN t}
  • !f s. FINITE s ==> FINITE {y | ?x. x IN s /\ y = f x}
  • !f s. FINITE s ==> FINITE (IMAGE f s)
  • !s x. FINITE (x INSERT s) <=> FINITE s
  • !s t u. s SUBSET t UNION u <=> s DIFF t SUBSET u
  • !s t. FINITE (s UNION t) <=> FINITE s /\ FINITE t
  • !s t. FINITE t /\ s SUBSET t ==> FINITE s
  • !s. FINITE s ==> (FINITE (UNIONS s) <=> (!t. t IN s ==> FINITE t))
    		•
    Constructive Lemmas
  • T
  • !x y s. x IN y INSERT s <=> x = y \/ x IN s
  • !x y. x = y <=> y = x
  • !x y. x = y ==> y = x
  • !x s t. s SUBSET x INSERT t <=> s DELETE x SUBSET t
  • !x s t. x INSERT s UNION t = x INSERT (s UNION t)
  • !x s. s DIFF {x} = s DELETE x
  • !x s. x INSERT s = {y | y = x \/ y IN s}
  • !x s. {x} UNION s = x INSERT s
  • !x s. FINITE s ==> FINITE (x INSERT s)
  • !x s. x IN s ==> x INSERT (s DELETE x) = s
  • !x. ~(x IN {})
  • !x. x = x
  • !x. x IN UNIV
  • !y s f. y IN IMAGE f s <=> (?x. y = f x /\ x IN s)
  • !p1 p2 q1 q2. (p1 ==> p2) /\ (q1 ==> q2) ==> p1 /\ q1 ==> p2 /\ q2
  • !p1 p2 q1 q2. (p1 ==> p2) /\ (q1 ==> q2) ==> p1 \/ q1 ==> p2 \/ q2
  • !p q r. p ==> q ==> r <=> p /\ q ==> r
  • !t1 t2 t3. (t1 \/ t2) \/ t3 <=> t1 \/ t2 \/ t3
  • !t1 t2. t1 /\ t2 <=> t2 /\ t1
  • !t1 t2. t1 \/ t2 <=> t2 \/ t1
  • !p q. (!x. p ==> q x) <=> p ==> (!x. q x)
  • !t. (!x. t) <=> t
  • !t. (?x. t) <=> t
  • !t. F /\ t <=> F
  • !t. T /\ t <=> t
  • !t. t /\ F <=> F
  • !t. t /\ T <=> t
  • !t. t /\ t <=> t
  • !t. (F <=> t) <=> ~t
  • !t. (T <=> t) <=> t
  • !t. (t <=> F) <=> ~t
  • !t. (t <=> T) <=> t
  • !t. F ==> t <=> T
  • !t. T ==> t <=> t
  • !t. t ==> F <=> ~t
  • !t. t ==> T <=> T
  • !t. F \/ t <=> t
  • !t. T \/ t <=> T
  • !t. t \/ F <=> t
  • !t. t \/ T <=> T
  • !t. t \/ t <=> t
  • !t. t ==> t
  • !t. (t <=> T) \/ (t <=> F)
  • !f y. (\x. f x) y = f y
  • !f g. (!x. f x = g x) <=> f = g
  • !f g. (!x. f x = g x) ==> f = g
  • !f s t. FINITE t /\ (!y. y IN t ==> FINITE {x | x IN s /\ f x = y}) ==> FINITE {x | x IN s /\ f x IN t}
  • !f s. IMAGE f s = {y | ?x. x IN s /\ y = f x}
  • !f s. FINITE s ==> FINITE {y | ?x. x IN s /\ y = f x}
  • !f s. FINITE s ==> FINITE (IMAGE f s)
  • !t. (\x. t x) = t
  • !p a. (?x. a = x /\ p x) <=> p a
  • !p x. x IN GSPEC p <=> p x
  • !p x. x IN {y | p y} <=> p x
  • !p q. (!x. p x ==> q x) ==> (?x. p x) ==> (?x. q x)
  • !p. (!x y. p x y) <=> (!y x. p x y)
  • !p f s. (!y. y IN IMAGE f s ==> p y) <=> (!x. x IN s ==> p (f x))
  • !p f s. (?y. y IN IMAGE f s /\ p y) <=> (?x. x IN s /\ p (f x))
  • !p. p {} /\ (!x s. p s ==> p (x INSERT s)) ==> (!a. FINITE a ==> p a)
  • !s x y. x IN s DELETE y <=> x IN s /\ ~(x = y)
  • !s x. FINITE (x INSERT s) <=> FINITE s
  • !s x. s DELETE x = {y | y IN s /\ ~(y = x)}
  • !s t x. x IN s DIFF t <=> x IN s /\ ~(x IN t)
  • !s t x. x IN s UNION t <=> x IN s \/ x IN t
  • !s t u. s SUBSET t UNION u <=> s DIFF t SUBSET u
  • !s t u. (s UNION t) UNION u = s UNION t UNION u
  • !s t. (!x. x IN s <=> x IN t) <=> s = t
  • !s t. FINITE (s UNION t) <=> FINITE s /\ FINITE t
  • !s t. s SUBSET t <=> (!x. x IN s ==> x IN t)
  • !s t. s DIFF t = {x | x IN s /\ ~(x IN t)}
  • !s t. s UNION t = {x | x IN s \/ x IN t}
  • !s t. s UNION t = t UNION s
  • !s t. (!x. x IN s <=> x IN t) ==> s = t
  • !s t. FINITE t /\ s SUBSET t ==> FINITE s
  • !s t. FINITE s /\ FINITE t ==> FINITE (s UNION t)
  • !s t. s SUBSET t UNION s
  • !s t. s SUBSET s UNION t
  • !s u. UNIONS (s INSERT u) = s UNION UNIONS u
  • !s. s SUBSET {} <=> s = {}
  • !s. {} UNION s = s
  • !s. s UNION {} = s
  • !s x. x IN UNIONS s <=> (?t. t IN s /\ x IN t)
  • !s. UNIONS s = {x | ?u. u IN s /\ x IN u}
  • !s. FINITE s ==> (FINITE (UNIONS s) <=> (!t. t IN s ==> FINITE t))
  • FINITE {}
  • F <=> (!p. p)
  • T <=> (\p. p) = (\p. p)
  • ~F <=> T
  • ~T <=> F
  • (~) = (\p. p ==> F)
  • (/\) = (\p q. (\f. f p q) = (\f. f T T))
  • (==>) = (\p q. p /\ q <=> p)
  • (\/) = (\p q. !r. (p ==> r) ==> (q ==> r) ==> r)
  • (!) = (\p. p = (\x. T))
  • (?) = (\p. !q. (!x. p x ==> q) ==> q)
  • {} = {x | F}
  • UNIV = {x | T}
  • GSPEC (\x. F) = {}
  • UNIONS {} = {}
    		•
    Contained Package set-finite-thm
    Comment Standard HOL library retrieved from OpenTheory
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