Theorem fvmpt2f

Description: Value of a function given by the maps-to notation. (Contributed by Thierry Arnoux, 9-Mar-2017)

	Ref	Expression
Hypothesis	fvmpt2f.0	$$⊢\underset{\_}{Ⅎ}xA$$
Assertion	fvmpt2f	$$⊢\left(x\in A\wedge B\in C\right)\to \left(x\in A⟼B\right)\left(x\right)=B$$

Proof

Step	Hyp	Ref	Expression
1		fvmpt2f.0	$$⊢\underset{\_}{Ⅎ}xA$$
2		csbeq1	$$⊢y=x\to ⦋y/x⦌B=⦋x/x⦌B$$
3		csbid	$$⊢⦋x/x⦌B=B$$
4	2 3	syl6eq	$$⊢y=x\to ⦋y/x⦌B=B$$
5		nfcv	$$⊢\underset{\_}{Ⅎ}yA$$
6		nfcv	$$⊢\underset{\_}{Ⅎ}yB$$
7		nfcsb1v	$$⊢\underset{\_}{Ⅎ}x⦋y/x⦌B$$
8		csbeq1a	$$⊢x=y\to B=⦋y/x⦌B$$
9	1 5 6 7 8	cbvmptf	$$⊢\left(x\in A⟼B\right)=\left(y\in A⟼⦋y/x⦌B\right)$$
10	4 9	fvmptg	$$⊢\left(x\in A\wedge B\in C\right)\to \left(x\in A⟼B\right)\left(x\right)=B$$