Theorem ixpeq1i 36771

Description: Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.)

Hypothesis

Ref	Expression
ixpeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
ixpeq1i	⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶

Proof of Theorem ixpeq1i

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ixpeq1i.1	. . . . . . 7 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2857	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3	2	abbii 2832	. . . . 5 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥 ∣ 𝑥 ∈ 𝐵}
4	3	fneq2i 6637	. . . 4 ⊢ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐵})
5	2	imbi1i 352	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → (𝑓‘𝑥) ∈ 𝐶))
6	5	ralbii2 3109	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)
7	4, 6	anbi12i 640	. . 3 ⊢ ((𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐵} ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶))
8	7	abbii 2832	. 2 ⊢ {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)} = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐵} ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)}
9		df-ixp 8902	. 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)}
10		df-ixp 8902	. 2 ⊢ X𝑥 ∈ 𝐵 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐵} ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)}
11	8, 9, 10	3eqtr4i 2798	1 ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2743  ∀wral 3081   Fn wfn 6535  ‘cfv 6540  Xcixp 8901

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-fn 6543  df-ixp 8902

This theorem is used by:  ixpeq12i  36772