Theorem mpoeq3ia

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

	Ref	Expression
Hypothesis	mpoeq3ia.1	$$⊢\left(x\in A\wedge y\in B\right)\to C=D$$
Assertion	mpoeq3ia	$$⊢\left(x\in A,y\in B⟼C\right)=\left(x\in A,y\in B⟼D\right)$$

Proof

Step	Hyp	Ref	Expression
1		mpoeq3ia.1	$$⊢\left(x\in A\wedge y\in B\right)\to C=D$$
2	1	3adant1	$$⊢\left(\top \wedge x\in A\wedge y\in B\right)\to C=D$$
3	2	mpoeq3dva	$$⊢\top \to \left(x\in A,y\in B⟼C\right)=\left(x\in A,y\in B⟼D\right)$$
4	3	mptru	$$⊢\left(x\in A,y\in B⟼C\right)=\left(x\in A,y\in B⟼D\right)$$