| 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 UNIVF <=> (!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 |
