Theorem cbvmptdavw

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

	Ref	Expression
Hypothesis	cbvmptdavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐵 = 𝐶 )
Assertion	cbvmptdavw	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑦 ∈ 𝐴 ↦ 𝐶 ) )

Proof

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