Theorem mpteq12df

Description: An equality inference for the maps-to notation. Compare mpteq12dv . (Contributed by Scott Fenton, 8-Aug-2013) (Revised by Mario Carneiro, 11-Dec-2016)

	Ref	Expression
Hypotheses	mpteq12df.1	⊢ Ⅎ 𝑥 𝜑
	mpteq12df.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
	mpteq12df.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
Assertion	mpteq12df	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		mpteq12df.1	⊢ Ⅎ 𝑥 𝜑
2		mpteq12df.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
3		mpteq12df.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
4		nfv	⊢ Ⅎ 𝑦 𝜑
5	2	eleq2d	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶 ) )
6	3	eqeq2d	⊢ ( 𝜑 → ( 𝑦 = 𝐵 ↔ 𝑦 = 𝐷 ) )
7	5 6	anbi12d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( 𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷 ) ) )
8	1 4 7	opabbid	⊢ ( 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 ) )