Theorem mpteq2ia

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013)

	Ref	Expression
Hypothesis	mpteq2ia.1	$$⊢x\in A\to B=C$$
Assertion	mpteq2ia	$$⊢\left(x\in A⟼B\right)=\left(x\in A⟼C\right)$$

Proof

Step	Hyp	Ref	Expression
1		mpteq2ia.1	$$⊢x\in A\to B=C$$
2		eqid	$$⊢A=A$$
3	2	ax-gen	$$⊢\forall xA=A$$
4	1	rgen	$$⊢\forall x\in AB=C$$
5		mpteq12f	$$⊢\left(\forall xA=A\wedge \forall x\in AB=C\right)\to \left(x\in A⟼B\right)=\left(x\in A⟼C\right)$$
6	3 4 5	mp2an	$$⊢\left(x\in A⟼B\right)=\left(x\in A⟼C\right)$$