Theorem mpteq12

Description: An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013)

	Ref	Expression
Assertion	mpteq12	$$⊢\left(A=C\wedge \forall x\in AB=D\right)\to \left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-5	$$⊢A=C\to \forall xA=C$$
2		mpteq12f	$$⊢\left(\forall xA=C\wedge \forall x\in AB=D\right)\to \left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$
3	1 2	sylan	$$⊢\left(A=C\wedge \forall x\in AB=D\right)\to \left(x\in A⟼B\right)=\left(x\in C⟼D\right)$$