Theorem cbvmptdavw2

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

	Ref	Expression
Hypotheses	cbvmptdavw2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐶 = 𝐷 )
	cbvmptdavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
Assertion	cbvmptdavw2	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑦 ∈ 𝐵 ↦ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvmptdavw2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐶 = 𝐷 )
2		cbvmptdavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
3		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
4	3	adantl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
5	2	eleq2d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
6	4 5	bitrd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
7	1	eqeq2d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑡 = 𝐶 ↔ 𝑡 = 𝐷 ) )
8	6 7	anbi12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷 ) ) )
9	8	cbvopab1davw	⊢ ( 𝜑 → { ⟨ 𝑥 , 𝑡 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶 ) } = { ⟨ 𝑦 , 𝑡 ⟩ ∣ ( 𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷 ) } )
10		df-mpt	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = { ⟨ 𝑥 , 𝑡 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶 ) }
11		df-mpt	⊢ ( 𝑦 ∈ 𝐵 ↦ 𝐷 ) = { ⟨ 𝑦 , 𝑡 ⟩ ∣ ( 𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷 ) }
12	9 10 11	3eqtr4g	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑦 ∈ 𝐵 ↦ 𝐷 ) )