Theorem mpteq2ia

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013) (Proof shortened by SN, 11-Nov-2024)

	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	1	adantl	$$⊢\left(\top \wedge x\in A\right)\to B=C$$
3	2	mpteq2dva	$$⊢\top \to \left(x\in A⟼B\right)=\left(x\in A⟼C\right)$$
4	3	mptru	$$⊢\left(x\in A⟼B\right)=\left(x\in A⟼C\right)$$