Theorem mpteq12da

Description: An equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021) Remove dependency on ax-10 . (Revised by SN, 11-Nov-2024)

	Ref	Expression
Hypotheses	mpteq12da.1	⊢ Ⅎ 𝑥 𝜑
	mpteq12da.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
	mpteq12da.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐷 )
Assertion	mpteq12da	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		mpteq12da.1	⊢ Ⅎ 𝑥 𝜑
2		mpteq12da.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
3		mpteq12da.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐷 )
4		nfv	⊢ Ⅎ 𝑦 𝜑
5	3	eqeq2d	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 = 𝐵 ↔ 𝑦 = 𝐷 ) )
6	5	pm5.32da	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ) )
7	2	eleq2d	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶 ) )
8	7	anbi1d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷 ) ) )
9	6 8	bitrd	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷 ) ) )
10	1 4 9	opabbid	⊢ ( 𝜑 → { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷 ) } )
11		df-mpt	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) }
12		df-mpt	⊢ ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷 ) }
13	10 11 12	3eqtr4g	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )