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Theorem ixpeq1i 36179
Description: Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.)
Hypothesis
Ref Expression
ixpeq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
ixpeq1i X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶

Proof of Theorem ixpeq1i
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ixpeq1i.1 . . . . . . 7 𝐴 = 𝐵
21eleq2i 2832 . . . . . 6 (𝑥𝐴𝑥𝐵)
32abbii 2808 . . . . 5 {𝑥𝑥𝐴} = {𝑥𝑥𝐵}
43fneq2i 6664 . . . 4 (𝑓 Fn {𝑥𝑥𝐴} ↔ 𝑓 Fn {𝑥𝑥𝐵})
52imbi1i 349 . . . . 5 ((𝑥𝐴 → (𝑓𝑥) ∈ 𝐶) ↔ (𝑥𝐵 → (𝑓𝑥) ∈ 𝐶))
65ralbii2 3088 . . . 4 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶 ↔ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)
74, 6anbi12i 628 . . 3 ((𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶) ↔ (𝑓 Fn {𝑥𝑥𝐵} ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶))
87abbii 2808 . 2 {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)} = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐵} ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)}
9 df-ixp 8934 . 2 X𝑥𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)}
10 df-ixp 8934 . 2 X𝑥𝐵 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐵} ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)}
118, 9, 103eqtr4i 2774 1 X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  {cab 2713  wral 3060   Fn wfn 6554  cfv 6559  Xcixp 8933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-fn 6562  df-ixp 8934
This theorem is referenced by:  ixpeq12i  36180
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