Metamath Proof Explorer


Theorem f1dom2gOLD

Description: Obsolete version of f1dom2g as of 25-Sep-2024. (Contributed by Mario Carneiro, 24-Jun-2015) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion f1dom2gOLD
|- ( ( 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 )