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Theorem ixpeq1 8949
Description: Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
Assertion
Ref Expression
ixpeq1 (𝐴 = 𝐵X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem ixpeq1
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 fneq2 6659 . . . 4 (𝐴 = 𝐵 → (𝑓 Fn 𝐴𝑓 Fn 𝐵))
2 raleq 3322 . . . 4 (𝐴 = 𝐵 → (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶 ↔ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶))
31, 2anbi12d 632 . . 3 (𝐴 = 𝐵 → ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐵 ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)))
43abbidv 2807 . 2 (𝐴 = 𝐵 → {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)} = {𝑓 ∣ (𝑓 Fn 𝐵 ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)})
5 dfixp 8940 . 2 X𝑥𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)}
6 dfixp 8940 . 2 X𝑥𝐵 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐵 ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)}
74, 5, 63eqtr4g 2801 1 (𝐴 = 𝐵X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  {cab 2713  wral 3060   Fn wfn 6555  cfv 6560  Xcixp 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-fn 6563  df-ixp 8939
This theorem is referenced by:  ixpeq1d  8950  finixpnum  37613  ioorrnopn  46325  ioorrnopnxr  46327  ovnval  46561  hoicvr  46568  hoidmv1le  46614  hoidmvle  46620  ovnhoi  46623  hspval  46629  ovnlecvr2  46630  hoiqssbl  46645  vonhoire  46692  iunhoiioo  46696  vonioo  46702  vonicc  46705
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