Theorem f1dom2g

Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015)

	Ref	Expression
Assertion	f1dom2g	|- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> A ~<_ B )

Proof

Step	Hyp	Ref	Expression
1		f1f	|- ( F : A -1-1-> B -> F : A --> B )
2		fex2	|- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V )
3	1 2	syl3an1	|- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) -> F e. _V )
4	3	3coml	|- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> F e. _V )
5		simp3	|- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> F : A -1-1-> B )
6		f1eq1	|- ( f = F -> ( f : A -1-1-> B <-> F : A -1-1-> B ) )
7	4 5 6	spcedv	|- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> E. f f : A -1-1-> B )
8		brdomg	|- ( B e. W -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
9	8	3ad2ant2	|- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
10	7 9	mpbird	|- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> A ~<_ B )