Theorem ixpeq1i 36508

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 2848	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3	2	abbii 2823	. . . . 5 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥 ∣ 𝑥 ∈ 𝐵}
4	3	fneq2i 6608	. . . 4 ⊢ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐵})
5	2	imbi1i 351	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → (𝑓‘𝑥) ∈ 𝐶))
6	5	ralbii2 3098	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)
7	4, 6	anbi12i 636	. . 3 ⊢ ((𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐵} ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶))
8	7	abbii 2823	. 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 8869	. 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)}
10		df-ixp 8869	. 2 ⊢ X𝑥 ∈ 𝐵 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐵} ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)}
11	8, 9, 10	3eqtr4i 2789	1 ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶

Syntax hints:   ∧ wa 398   = wceq 1554   ∈ wcel 2136  {cab 2734  ∀wral 3070   Fn wfn 6505  ‘cfv 6510  Xcixp 8868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-fn 6513  df-ixp 8869

This theorem is referenced by:  ixpeq12i  36509