Theorem fvmpt2df

Description: Deduction version of fvmpt2 . (Contributed by Glauco Siliprandi, 24-Jan-2025)

	Ref	Expression
Hypotheses	fvmpt2df.1	$$⊢\underset{\_}{Ⅎ}xA$$
	fvmpt2df.2	$$⊢F=\left(x\in A⟼B\right)$$
	fvmpt2df.3	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
Assertion	fvmpt2df	$$⊢\left(\phi \wedge x\in A\right)\to F\left(x\right)=B$$

Proof

Step	Hyp	Ref	Expression
1		fvmpt2df.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		fvmpt2df.2	$$⊢F=\left(x\in A⟼B\right)$$
3		fvmpt2df.3	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
4	2	fveq1i	$$⊢F\left(x\right)=\left(x\in A⟼B\right)\left(x\right)$$
5		id	$$⊢x\in A\to x\in A$$
6	1	fvmpt2f	$$⊢\left(x\in A\wedge B\in V\right)\to \left(x\in A⟼B\right)\left(x\right)=B$$
7	5 3 6	syl2an2	$$⊢\left(\phi \wedge x\in A\right)\to \left(x\in A⟼B\right)\left(x\right)=B$$
8	4 7	eqtrid	$$⊢\left(\phi \wedge x\in A\right)\to F\left(x\right)=B$$