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Metamath Proof Explorer

Theorem mpteq12dva

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017) Remove dependency on ax-10 , ax-12 . (Revised by SN, 11-Nov-2024)

		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	2	eqeq2d	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 = 𝐵 ↔ 𝑦 = 𝐷 ) )
4	3	pm5.32da	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ) )
5	1	eleq2d	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶 ) )
6	5	anbi1d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷 ) ) )
7	4 6	bitrd	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷 ) ) )
8	7	opabbidv	⊢ ( 𝜑 → { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷 ) } )
9		df-mpt	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) }
10		df-mpt	⊢ ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷 ) }
11	8 9 10	3eqtr4g	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )