Theorem fvmpt2i

Description: Value of a function given by the maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014)

	Ref	Expression
Hypothesis	mptrcl.1	$$⊢F=\left(x\in A⟼B\right)$$
Assertion	fvmpt2i	$$⊢x\in A\to F\left(x\right)=\mathrm{I}\left(B\right)$$

Proof

Step	Hyp	Ref	Expression
1		mptrcl.1	$$⊢F=\left(x\in A⟼B\right)$$
2		csbeq1	$$⊢y=x\to ⦋y/x⦌B=⦋x/x⦌B$$
3		csbid	$$⊢⦋x/x⦌B=B$$
4	2 3	eqtrdi	$$⊢y=x\to ⦋y/x⦌B=B$$
5		nfcv	$$⊢\underset{\_}{Ⅎ}yB$$
6		nfcsb1v	$$⊢\underset{\_}{Ⅎ}x⦋y/x⦌B$$
7		csbeq1a	$$⊢x=y\to B=⦋y/x⦌B$$
8	5 6 7	cbvmpt	$$⊢\left(x\in A⟼B\right)=\left(y\in A⟼⦋y/x⦌B\right)$$
9	1 8	eqtri	$$⊢F=\left(y\in A⟼⦋y/x⦌B\right)$$
10	4 9	fvmpti	$$⊢x\in A\to F\left(x\right)=\mathrm{I}\left(B\right)$$