Theorem fvmpt2d

Description: Deduction version of fvmpt2 . (Contributed by Thierry Arnoux, 8-Dec-2016)

	Ref	Expression
Hypotheses	fvmpt2d.1	|- ( ph -> F = ( x e. A |-> B ) )
	fvmpt2d.4	|- ( ( ph /\ x e. A ) -> B e. V )
Assertion	fvmpt2d	|- ( ( ph /\ x e. A ) -> ( F ` x ) = B )

Proof

Step	Hyp	Ref	Expression
1		fvmpt2d.1	|- ( ph -> F = ( x e. A |-> B ) )
2		fvmpt2d.4	|- ( ( ph /\ x e. A ) -> B e. V )
3	1	fveq1d	|- ( ph -> ( F ` x ) = ( ( x e. A |-> B ) ` x ) )
4	3	adantr	|- ( ( ph /\ x e. A ) -> ( F ` x ) = ( ( x e. A |-> B ) ` x ) )
5		id	|- ( x e. A -> x e. A )
6		eqid	|- ( x e. A |-> B ) = ( x e. A |-> B )
7	6	fvmpt2	|- ( ( x e. A /\ B e. V ) -> ( ( x e. A |-> B ) ` x ) = B )
8	5 2 7	syl2an2	|- ( ( ph /\ x e. A ) -> ( ( x e. A |-> B ) ` x ) = B )
9	4 8	eqtrd	|- ( ( ph /\ x e. A ) -> ( F ` x ) = B )