Theorem cbvixpvw2

Description: Change bound variable and domain in an indexed Cartesian product, using implicit substitution. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvixpvw2.1	⊢ ( 𝑥 = 𝑦 → 𝐶 = 𝐷 )
	cbvixpvw2.2	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
Assertion	cbvixpvw2	⊢ X 𝑥 ∈ 𝐴 𝐶 = X 𝑦 ∈ 𝐵 𝐷

Proof

Step	Hyp	Ref	Expression
1		cbvixpvw2.1	⊢ ( 𝑥 = 𝑦 → 𝐶 = 𝐷 )
2		cbvixpvw2.2	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
3		id	⊢ ( 𝑥 = 𝑦 → 𝑥 = 𝑦 )
4	3 2	eleq12d	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
5	4	cbvabv	⊢ { 𝑥 ∣ 𝑥 ∈ 𝐴 } = { 𝑦 ∣ 𝑦 ∈ 𝐵 }
6	5	fneq2i	⊢ ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ↔ 𝑡 Fn { 𝑦 ∣ 𝑦 ∈ 𝐵 } )
7		fveq2	⊢ ( 𝑥 = 𝑦 → ( 𝑡 ‘ 𝑥 ) = ( 𝑡 ‘ 𝑦 ) )
8	7 1	eleq12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ↔ ( 𝑡 ‘ 𝑦 ) ∈ 𝐷 ) )
9	2 8	cbvralvw2	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ↔ ∀ 𝑦 ∈ 𝐵 ( 𝑡 ‘ 𝑦 ) ∈ 𝐷 )
10	6 9	anbi12i	⊢ ( ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) ↔ ( 𝑡 Fn { 𝑦 ∣ 𝑦 ∈ 𝐵 } ∧ ∀ 𝑦 ∈ 𝐵 ( 𝑡 ‘ 𝑦 ) ∈ 𝐷 ) )
11	10	abbii	⊢ { 𝑡 ∣ ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) } = { 𝑡 ∣ ( 𝑡 Fn { 𝑦 ∣ 𝑦 ∈ 𝐵 } ∧ ∀ 𝑦 ∈ 𝐵 ( 𝑡 ‘ 𝑦 ) ∈ 𝐷 ) }
12		df-ixp	⊢ X 𝑥 ∈ 𝐴 𝐶 = { 𝑡 ∣ ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) }
13		df-ixp	⊢ X 𝑦 ∈ 𝐵 𝐷 = { 𝑡 ∣ ( 𝑡 Fn { 𝑦 ∣ 𝑦 ∈ 𝐵 } ∧ ∀ 𝑦 ∈ 𝐵 ( 𝑡 ‘ 𝑦 ) ∈ 𝐷 ) }
14	11 12 13	3eqtr4i	⊢ X 𝑥 ∈ 𝐴 𝐶 = X 𝑦 ∈ 𝐵 𝐷