Theorem cbvmptvw2

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

	Ref	Expression
Hypotheses	cbvmptvw2.1	⊢ ( 𝑥 = 𝑦 → 𝐶 = 𝐷 )
	cbvmptvw2.2	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
Assertion	cbvmptvw2	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑦 ∈ 𝐵 ↦ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		cbvmptvw2.1	⊢ ( 𝑥 = 𝑦 → 𝐶 = 𝐷 )
2		cbvmptvw2.2	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
3		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
4	2	eleq2d	⊢ ( 𝑥 = 𝑦 → ( 𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
5	3 4	bitrd	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
6	1	eqeq2d	⊢ ( 𝑥 = 𝑦 → ( 𝑡 = 𝐶 ↔ 𝑡 = 𝐷 ) )
7	5 6	anbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷 ) ) )
8	7	cbvopab1v	⊢ { ⟨ 𝑥 , 𝑡 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶 ) } = { ⟨ 𝑦 , 𝑡 ⟩ ∣ ( 𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷 ) }
9		df-mpt	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = { ⟨ 𝑥 , 𝑡 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶 ) }
10		df-mpt	⊢ ( 𝑦 ∈ 𝐵 ↦ 𝐷 ) = { ⟨ 𝑦 , 𝑡 ⟩ ∣ ( 𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷 ) }
11	8 9 10	3eqtr4i	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑦 ∈ 𝐵 ↦ 𝐷 )