Metamath Proof Explorer

Theorem fnfvof

Description: Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014) (Proof shortened by Mario Carneiro, 5-Jun-2015)

Ref Expression
Assertion fnfvof 
|- ( ( ( F Fn A /\ G Fn A ) /\ ( A e. V /\ X e. A ) ) -> ( ( F oF R G ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )

Proof

Step Hyp Ref Expression
1 simpll 
 |-  ( ( ( F Fn A /\ G Fn A ) /\ A e. V ) -> F Fn A )
2 simplr 
 |-  ( ( ( F Fn A /\ G Fn A ) /\ A e. V ) -> G Fn A )
3 simpr 
 |-  ( ( ( F Fn A /\ G Fn A ) /\ A e. V ) -> A e. V )
4 inidm 
 |-  ( A i^i A ) = A
5 eqidd 
 |-  ( ( ( ( F Fn A /\ G Fn A ) /\ A e. V ) /\ X e. A ) -> ( F ` X ) = ( F ` X ) )
6 eqidd 
 |-  ( ( ( ( F Fn A /\ G Fn A ) /\ A e. V ) /\ X e. A ) -> ( G ` X ) = ( G ` X ) )
7 1 2 3 3 4 5 6 ofval 
 |-  ( ( ( ( F Fn A /\ G Fn A ) /\ A e. V ) /\ X e. A ) -> ( ( F oF R G ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )
8 7 anasss 
 |-  ( ( ( F Fn A /\ G Fn A ) /\ ( A e. V /\ X e. A ) ) -> ( ( F oF R G ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )