Entry Value
Name INJECTIVE_IMAGE
Conclusion !f. (!s t. IMAGE f s = IMAGE f t ==> s = t) <=> (!x y. f x = f y ==> x = y)
Constructive Proof No
Axiom
!t. t \/ ~t 
(\a. a = (\b. (\c. c) = (\c. c))) (\d. (\e. d e) = d)
Classical Lemmas
  • !t. ~ ~t <=> t
  • !t. (t <=> T) \/ (t <=> F)
  • !f u. (!s t. s SUBSET u /\ t SUBSET u /\ IMAGE f s = IMAGE f t ==> s = t) <=> (!x y. x IN u /\ y IN u /\ f x = f y ==> x = y)
  • 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 IN {y} <=> x = y
  • !x y. x = y ==> y = x
  • !x s. x INSERT s = {y | y = x \/ y IN 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)
  • !t1 t2. t1 /\ t2 <=> t2 /\ t1
  • !t. (!x. t) <=> t
  • !t. ~ ~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 u. (!s t. s SUBSET u /\ t SUBSET u /\ IMAGE f s = IMAGE f t ==> s = t) <=> (!x y. x IN u /\ y IN u /\ f x = f y ==> x = y)
  • !f s. IMAGE f s = {y | ?x. x IN s /\ y = f x}
  • !t. (\x. t x) = t
  • !p a. (?x. x = a /\ p x) <=> p a
  • !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
  • !s t. (!x. x IN s <=> x IN t) <=> s = t
  • !s t. s SUBSET t <=> (!x. x IN s ==> x IN t)
  • !s t. (!x. x IN s <=> x IN t) ==> s = t
  • !s. s SUBSET UNIV
  • 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}
  • Contained Package set-thm
    Comment Standard HOL library retrieved from OpenTheory
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