FINITE_FINITE_PREIMAGE

Entry	Value
Name	FINITE_FINITE_PREIMAGE_GENERAL
Conclusion	!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}
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. 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 !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. 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} GSPEC (\x. F) = {} UNIONS {} = {}
Contained Package	set-finite-thm
Comment	Standard HOL library retrieved from OpenTheory

Back to main package page Back to contained package page