Theorem mpteq12dva

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017)

	Ref	Expression
Hypotheses	mpteq12dv.1	⊢ ( 𝜑 → 𝐴 = 𝐶 )
	mpteq12dva.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐷 )
Assertion	mpteq12dva	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		mpteq12dv.1	⊢ ( 𝜑 → 𝐴 = 𝐶 )
2		mpteq12dva.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐷 )
3	1	alrimiv	⊢ ( 𝜑 → ∀ 𝑥 𝐴 = 𝐶 )
4	2	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐷 )
5		mpteq12f	⊢ ( ( ∀ 𝑥 𝐴 = 𝐶 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐷 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )
6	3 4 5	syl2anc	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )