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Metamath Proof Explorer

Theorem f1oabexg

Description: The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008) (Proof shortened by AV, 9-Jun-2025)

Ref Expression
Hypothesis f1oabexg.1 
|- F = { f | ( f : A -1-1-onto-> B /\ ph ) }
Assertion f1oabexg 
|- ( ( 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 elex 
 |-  ( A e. C -> A e. _V )
3 elex 
 |-  ( B e. D -> B e. _V )
4 f1of 
 |-  ( f : A -1-1-onto-> B -> f : A --> B )
5 4 ad2antrl 
 |-  ( ( ( A e. _V /\ B e. _V ) /\ ( f : A -1-1-onto-> B /\ ph ) ) -> f : A --> B )
6 simpl 
 |-  ( ( A e. _V /\ B e. _V ) -> A e. _V )
7 simpr 
 |-  ( ( A e. _V /\ B e. _V ) -> B e. _V )
8 5 6 7 fabexd 
 |-  ( ( A e. _V /\ B e. _V ) -> { f | ( f : A -1-1-onto-> B /\ ph ) } e. _V )
9 1 8 eqeltrid 
 |-  ( ( A e. _V /\ B e. _V ) -> F e. _V )
10 2 3 9 syl2an 
 |-  ( ( A e. C /\ B e. D ) -> F e. _V )