Theorem fvmpt

Description: Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011)

	Ref	Expression
Hypotheses	fvmptg.1	$$⊢x=A\to B=C$$
	fvmptg.2	$$⊢F=\left(x\in D⟼B\right)$$
	fvmpt.3	$$⊢C\in \mathrm{V}$$
Assertion	fvmpt	$$⊢A\in D\to F\left(A\right)=C$$

Proof

Step	Hyp	Ref	Expression
1		fvmptg.1	$$⊢x=A\to B=C$$
2		fvmptg.2	$$⊢F=\left(x\in D⟼B\right)$$
3		fvmpt.3	$$⊢C\in \mathrm{V}$$
4	1 2	fvmptg	$$⊢\left(A\in D\wedge C\in \mathrm{V}\right)\to F\left(A\right)=C$$
5	3 4	mpan2	$$⊢A\in D\to F\left(A\right)=C$$