Theorem fvmpt4d

Description: Value of a function given by the maps-to notation. (Contributed by Glauco Siliprandi, 15-Feb-2025)

	Ref	Expression
Hypotheses	fvmpt4d.1	$$⊢\underset{\_}{Ⅎ}xA$$
	fvmpt4d.2	$$⊢\phi \to B\in C$$
	fvmpt4d.3	$$⊢\phi \to x\in A$$
Assertion	fvmpt4d	$$⊢\phi \to \left(x\in A⟼B\right)\left(x\right)=B$$

Proof

Step	Hyp	Ref	Expression
1		fvmpt4d.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		fvmpt4d.2	$$⊢\phi \to B\in C$$
3		fvmpt4d.3	$$⊢\phi \to x\in A$$
4	1	fvmpt2f	$$⊢\left(x\in A\wedge B\in C\right)\to \left(x\in A⟼B\right)\left(x\right)=B$$
5	3 2 4	syl2anc	$$⊢\phi \to \left(x\in A⟼B\right)\left(x\right)=B$$