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Metamath Proof Explorer

Theorem mpteq12dv

Description: An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011) (Revised by Mario Carneiro, 16-Dec-2013) Remove dependency on ax-10 , ax-12 . (Revised by SN and GG, 1-Dec-2023)

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

Proof

Step	Hyp	Ref	Expression
1		mpteq12dv.1	⊢ ( 𝜑 → 𝐴 = 𝐶 )
2		mpteq12dv.2	⊢ ( 𝜑 → 𝐵 = 𝐷 )
3	2	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐷 )
4	1 3	mpteq12dva	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )