Theorem nfmpt

Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013)

	Ref	Expression
Hypotheses	nfmpt.1	$$⊢\underset{\_}{Ⅎ}xA$$
	nfmpt.2	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	nfmpt	$$⊢\underset{\_}{Ⅎ}x\left(y\in A⟼B\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfmpt.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfmpt.2	$$⊢\underset{\_}{Ⅎ}xB$$
3		df-mpt	$$⊢\left(y\in A⟼B\right)=\left\{⟨y,z⟩|\left(y\in A\wedge z=B\right)\right\}$$
4	1	nfcri	$$⊢Ⅎxy\in A$$
5	2	nfeq2	$$⊢Ⅎxz=B$$
6	4 5	nfan	$$⊢Ⅎx\left(y\in A\wedge z=B\right)$$
7	6	nfopab	$$⊢\underset{\_}{Ⅎ}x\left\{⟨y,z⟩|\left(y\in A\wedge z=B\right)\right\}$$
8	3 7	nfcxfr	$$⊢\underset{\_}{Ⅎ}x\left(y\in A⟼B\right)$$