Theorem cbvixpdavw2

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

	Ref	Expression
Hypotheses	cbvixpdavw2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐶 = 𝐷 )
	cbvixpdavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
Assertion	cbvixpdavw2	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐶 = X 𝑦 ∈ 𝐵 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		cbvixpdavw2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐶 = 𝐷 )
2		cbvixpdavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
3		simpr	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝑥 = 𝑦 )
4	3 2	eleq12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
5	4	cbvabdavw	⊢ ( 𝜑 → { 𝑥 ∣ 𝑥 ∈ 𝐴 } = { 𝑦 ∣ 𝑦 ∈ 𝐵 } )
6	5	fneq2d	⊢ ( 𝜑 → ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ↔ 𝑡 Fn { 𝑦 ∣ 𝑦 ∈ 𝐵 } ) )
7		fveq2	⊢ ( 𝑥 = 𝑦 → ( 𝑡 ‘ 𝑥 ) = ( 𝑡 ‘ 𝑦 ) )
8	7	adantl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑡 ‘ 𝑥 ) = ( 𝑡 ‘ 𝑦 ) )
9	8 1	eleq12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ↔ ( 𝑡 ‘ 𝑦 ) ∈ 𝐷 ) )
10	9 2	cbvraldva2	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ↔ ∀ 𝑦 ∈ 𝐵 ( 𝑡 ‘ 𝑦 ) ∈ 𝐷 ) )
11	6 10	anbi12d	⊢ ( 𝜑 → ( ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) ↔ ( 𝑡 Fn { 𝑦 ∣ 𝑦 ∈ 𝐵 } ∧ ∀ 𝑦 ∈ 𝐵 ( 𝑡 ‘ 𝑦 ) ∈ 𝐷 ) ) )
12	11	abbidv	⊢ ( 𝜑 → { 𝑡 ∣ ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) } = { 𝑡 ∣ ( 𝑡 Fn { 𝑦 ∣ 𝑦 ∈ 𝐵 } ∧ ∀ 𝑦 ∈ 𝐵 ( 𝑡 ‘ 𝑦 ) ∈ 𝐷 ) } )
13		df-ixp	⊢ X 𝑥 ∈ 𝐴 𝐶 = { 𝑡 ∣ ( 𝑡 Fn { 𝑥 ∣ 𝑥 ∈ 𝐴 } ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑡 ‘ 𝑥 ) ∈ 𝐶 ) }
14		df-ixp	⊢ X 𝑦 ∈ 𝐵 𝐷 = { 𝑡 ∣ ( 𝑡 Fn { 𝑦 ∣ 𝑦 ∈ 𝐵 } ∧ ∀ 𝑦 ∈ 𝐵 ( 𝑡 ‘ 𝑦 ) ∈ 𝐷 ) }
15	12 13 14	3eqtr4g	⊢ ( 𝜑 → X 𝑥 ∈ 𝐴 𝐶 = X 𝑦 ∈ 𝐵 𝐷 )