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Metamath Proof Explorer

Theorem fabexg

Description: Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008) (Proof shortened by AV, 9-Jun-2025)

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

Proof

Step Hyp Ref Expression
1 fabexg.1 
 |-  F = { x | ( x : A --> B /\ ph ) }
2 elex 
 |-  ( A e. C -> A e. _V )
3 elex 
 |-  ( B e. D -> B e. _V )
4 simprl 
 |-  ( ( ( A e. _V /\ B e. _V ) /\ ( x : A --> B /\ ph ) ) -> x : A --> B )
5 simpl 
 |-  ( ( A e. _V /\ B e. _V ) -> A e. _V )
6 simpr 
 |-  ( ( A e. _V /\ B e. _V ) -> B e. _V )
7 4 5 6 fabexd 
 |-  ( ( A e. _V /\ B e. _V ) -> { x | ( x : A --> B /\ ph ) } e. _V )
8 1 7 eqeltrid 
 |-  ( ( A e. _V /\ B e. _V ) -> F e. _V )
9 2 3 8 syl2an 
 |-  ( ( A e. C /\ B e. D ) -> F e. _V )