Theorem cbvmpo1vw2

Description: Change domains and the first bound variable in a maps-to function, using implicit substitution. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvmpo1vw2.1	⊢ ( 𝑥 = 𝑧 → 𝐸 = 𝐹 )
	cbvmpo1vw2.2	⊢ ( 𝑥 = 𝑧 → 𝐶 = 𝐷 )
	cbvmpo1vw2.3	⊢ ( 𝑥 = 𝑧 → 𝐴 = 𝐵 )
Assertion	cbvmpo1vw2	⊢ ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐶 ↦ 𝐸 ) = ( 𝑧 ∈ 𝐵 , 𝑦 ∈ 𝐷 ↦ 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		cbvmpo1vw2.1	⊢ ( 𝑥 = 𝑧 → 𝐸 = 𝐹 )
2		cbvmpo1vw2.2	⊢ ( 𝑥 = 𝑧 → 𝐶 = 𝐷 )
3		cbvmpo1vw2.3	⊢ ( 𝑥 = 𝑧 → 𝐴 = 𝐵 )
4		id	⊢ ( 𝑥 = 𝑧 → 𝑥 = 𝑧 )
5	4 3	eleq12d	⊢ ( 𝑥 = 𝑧 → ( 𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵 ) )
6	2	eleq2d	⊢ ( 𝑥 = 𝑧 → ( 𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷 ) )
7	5 6	anbi12d	⊢ ( 𝑥 = 𝑧 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ↔ ( 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷 ) ) )
8	1	eqeq2d	⊢ ( 𝑥 = 𝑧 → ( 𝑡 = 𝐸 ↔ 𝑡 = 𝐹 ) )
9	7 8	anbi12d	⊢ ( 𝑥 = 𝑧 → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ∧ 𝑡 = 𝐸 ) ↔ ( ( 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑡 = 𝐹 ) ) )
10	9	cbvoprab1vw	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑡 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ∧ 𝑡 = 𝐸 ) } = { ⟨ ⟨ 𝑧 , 𝑦 ⟩ , 𝑡 ⟩ ∣ ( ( 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑡 = 𝐹 ) }
11		df-mpo	⊢ ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐶 ↦ 𝐸 ) = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑡 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ∧ 𝑡 = 𝐸 ) }
12		df-mpo	⊢ ( 𝑧 ∈ 𝐵 , 𝑦 ∈ 𝐷 ↦ 𝐹 ) = { ⟨ ⟨ 𝑧 , 𝑦 ⟩ , 𝑡 ⟩ ∣ ( ( 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑡 = 𝐹 ) }
13	10 11 12	3eqtr4i	⊢ ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐶 ↦ 𝐸 ) = ( 𝑧 ∈ 𝐵 , 𝑦 ∈ 𝐷 ↦ 𝐹 )