Theorem cbvmpodavw2

Description: Change bound variable and domains in a maps-to function. Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvmpodavw2.1	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → 𝐸 = 𝐹 )
	cbvmpodavw2.2	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → 𝐶 = 𝐷 )
	cbvmpodavw2.3	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → 𝐴 = 𝐵 )
Assertion	cbvmpodavw2	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐶 ↦ 𝐸 ) = ( 𝑧 ∈ 𝐵 , 𝑤 ∈ 𝐷 ↦ 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvmpodavw2.1	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → 𝐸 = 𝐹 )
2		cbvmpodavw2.2	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → 𝐶 = 𝐷 )
3		cbvmpodavw2.3	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → 𝐴 = 𝐵 )
4		simplr	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → 𝑥 = 𝑧 )
5	4 3	eleq12d	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵 ) )
6		simpr	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → 𝑦 = 𝑤 )
7	6 2	eleq12d	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → ( 𝑦 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷 ) )
8	5 7	anbi12d	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ↔ ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷 ) ) )
9	1	eqeq2d	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → ( 𝑡 = 𝐸 ↔ 𝑡 = 𝐹 ) )
10	8 9	anbi12d	⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑧 ) ∧ 𝑦 = 𝑤 ) → ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ∧ 𝑡 = 𝐸 ) ↔ ( ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷 ) ∧ 𝑡 = 𝐹 ) ) )
11	10	cbvoprab12davw	⊢ ( 𝜑 → { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑡 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ∧ 𝑡 = 𝐸 ) } = { ⟨ ⟨ 𝑧 , 𝑤 ⟩ , 𝑡 ⟩ ∣ ( ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷 ) ∧ 𝑡 = 𝐹 ) } )
12		df-mpo	⊢ ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐶 ↦ 𝐸 ) = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑡 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ∧ 𝑡 = 𝐸 ) }
13		df-mpo	⊢ ( 𝑧 ∈ 𝐵 , 𝑤 ∈ 𝐷 ↦ 𝐹 ) = { ⟨ ⟨ 𝑧 , 𝑤 ⟩ , 𝑡 ⟩ ∣ ( ( 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷 ) ∧ 𝑡 = 𝐹 ) }
14	11 12 13	3eqtr4g	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐶 ↦ 𝐸 ) = ( 𝑧 ∈ 𝐵 , 𝑤 ∈ 𝐷 ↦ 𝐹 ) )