Metamath Proof Explorer


Theorem f1oabexgOLD

Description: Obsolete version of f1oabexg as of 9-Jun-2025. (Contributed by Paul Chapman, 25-Feb-2008) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypothesis f1oabexg.1
|- F = { f | ( f : A -1-1-onto-> B /\ ph ) }
Assertion f1oabexgOLD
|- ( ( A e. C /\ B e. D ) -> F e. _V )

Proof

Step Hyp Ref Expression
1 f1oabexg.1
 |-  F = { f | ( f : A -1-1-onto-> B /\ ph ) }
2 f1of
 |-  ( f : A -1-1-onto-> B -> f : A --> B )
3 2 anim1i
 |-  ( ( f : A -1-1-onto-> B /\ ph ) -> ( f : A --> B /\ ph ) )
4 3 ss2abi
 |-  { f | ( f : A -1-1-onto-> B /\ ph ) } C_ { f | ( f : A --> B /\ ph ) }
5 eqid
 |-  { f | ( f : A --> B /\ ph ) } = { f | ( f : A --> B /\ ph ) }
6 5 fabexg
 |-  ( ( A e. C /\ B e. D ) -> { f | ( f : A --> B /\ ph ) } e. _V )
7 ssexg
 |-  ( ( { f | ( f : A -1-1-onto-> B /\ ph ) } C_ { f | ( f : A --> B /\ ph ) } /\ { f | ( f : A --> B /\ ph ) } e. _V ) -> { f | ( f : A -1-1-onto-> B /\ ph ) } e. _V )
8 4 6 7 sylancr
 |-  ( ( A e. C /\ B e. D ) -> { f | ( f : A -1-1-onto-> B /\ ph ) } e. _V )
9 1 8 eqeltrid
 |-  ( ( A e. C /\ B e. D ) -> F e. _V )