Theorem fvmpt2d

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

	Ref	Expression
Hypotheses	fvmpt2d.1	$$⊢\phi \to F=\left(x\in A⟼B\right)$$
	fvmpt2d.4	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
Assertion	fvmpt2d	$$⊢\left(\phi \wedge x\in A\right)\to F\left(x\right)=B$$

Proof

Step	Hyp	Ref	Expression
1		fvmpt2d.1	$$⊢\phi \to F=\left(x\in A⟼B\right)$$
2		fvmpt2d.4	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
3	1	fveq1d	$$⊢\phi \to F\left(x\right)=\left(x\in A⟼B\right)\left(x\right)$$
4	3	adantr	$$⊢\left(\phi \wedge x\in A\right)\to F\left(x\right)=\left(x\in A⟼B\right)\left(x\right)$$
5		id	$$⊢x\in A\to x\in A$$
6		eqid	$$⊢\left(x\in A⟼B\right)=\left(x\in A⟼B\right)$$
7	6	fvmpt2	$$⊢\left(x\in A\wedge B\in V\right)\to \left(x\in A⟼B\right)\left(x\right)=B$$
8	5 2 7	syl2an2	$$⊢\left(\phi \wedge x\in A\right)\to \left(x\in A⟼B\right)\left(x\right)=B$$
9	4 8	eqtrd	$$⊢\left(\phi \wedge x\in A\right)\to F\left(x\right)=B$$