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Theorem ixpeq1 8518
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 6430 . . . 4 (𝐴 = 𝐵 → (𝑓 Fn 𝐴𝑓 Fn 𝐵))
2 raleq 3310 . . . 4 (𝐴 = 𝐵 → (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶 ↔ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶))
31, 2anbi12d 634 . . 3 (𝐴 = 𝐵 → ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐵 ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)))
43abbidv 2802 . 2 (𝐴 = 𝐵 → {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)} = {𝑓 ∣ (𝑓 Fn 𝐵 ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)})
5 dfixp 8509 . 2 X𝑥𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)}
6 dfixp 8509 . 2 X𝑥𝐵 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐵 ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)}
74, 5, 63eqtr4g 2798 1 (𝐴 = 𝐵X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  {cab 2716  wral 3053   Fn wfn 6334  cfv 6339  Xcixp 8507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-fn 6342  df-ixp 8508
This theorem is referenced by:  ixpeq1d  8519  finixpnum  35385  ioorrnopn  43388  ioorrnopnxr  43390  ovnval  43621  hoicvr  43628  hoidmv1le  43674  hoidmvle  43680  ovnhoi  43683  hspval  43689  ovnlecvr2  43690  hoiqssbl  43705  vonhoire  43752  iunhoiioo  43756  vonioo  43762  vonicc  43765
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