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 ) ) )