Metamath Proof Explorer


Theorem ofcfval2

Description: The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 31-Jan-2017)

Ref Expression
Hypotheses ofcfval2.1
|- ( ph -> A e. V )
ofcfval2.2
|- ( ph -> C e. W )
ofcfval2.3
|- ( ( ph /\ x e. A ) -> B e. X )
ofcfval2.4
|- ( ph -> F = ( x e. A |-> B ) )
Assertion ofcfval2
|- ( ph -> ( F oFC R C ) = ( x e. A |-> ( B R C ) ) )

Proof

Step Hyp Ref Expression
1 ofcfval2.1
 |-  ( ph -> A e. V )
2 ofcfval2.2
 |-  ( ph -> C e. W )
3 ofcfval2.3
 |-  ( ( ph /\ x e. A ) -> B e. X )
4 ofcfval2.4
 |-  ( ph -> F = ( x e. A |-> B ) )
5 3 ralrimiva
 |-  ( ph -> A. x e. A B e. X )
6 eqid
 |-  ( x e. A |-> B ) = ( x e. A |-> B )
7 6 fnmpt
 |-  ( A. x e. A B e. X -> ( x e. A |-> B ) Fn A )
8 5 7 syl
 |-  ( ph -> ( x e. A |-> B ) Fn A )
9 4 fneq1d
 |-  ( ph -> ( F Fn A <-> ( x e. A |-> B ) Fn A ) )
10 8 9 mpbird
 |-  ( ph -> F Fn A )
11 4 3 fvmpt2d
 |-  ( ( ph /\ x e. A ) -> ( F ` x ) = B )
12 10 1 2 11 ofcfval
 |-  ( ph -> ( F oFC R C ) = ( x e. A |-> ( B R C ) ) )