Theorem f1oen3g

Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007) (Revised by Mario Carneiro, 10-Sep-2015)

	Ref	Expression
Assertion	f1oen3g	|- ( ( F e. V /\ F : A -1-1-onto-> B ) -> A ~~ B )

Proof

Step	Hyp	Ref	Expression
1		f1oeq1	|- ( f = F -> ( f : A -1-1-onto-> B <-> F : A -1-1-onto-> B ) )
2	1	spcegv	|- ( F e. V -> ( F : A -1-1-onto-> B -> E. f f : A -1-1-onto-> B ) )
3	2	imp	|- ( ( F e. V /\ F : A -1-1-onto-> B ) -> E. f f : A -1-1-onto-> B )
4		bren	|- ( A ~~ B <-> E. f f : A -1-1-onto-> B )
5	3 4	sylibr	|- ( ( F e. V /\ F : A -1-1-onto-> B ) -> A ~~ B )