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Theorem ixpeq1 8083
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 6139 . . . 4 (𝐴 = 𝐵 → (𝑓 Fn 𝐴𝑓 Fn 𝐵))
2 raleq 3275 . . . 4 (𝐴 = 𝐵 → (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶 ↔ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶))
31, 2anbi12d 749 . . 3 (𝐴 = 𝐵 → ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐵 ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)))
43abbidv 2877 . 2 (𝐴 = 𝐵 → {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)} = {𝑓 ∣ (𝑓 Fn 𝐵 ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)})
5 dfixp 8074 . 2 X𝑥𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)}
6 dfixp 8074 . 2 X𝑥𝐵 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐵 ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)}
74, 5, 63eqtr4g 2817 1 (𝐴 = 𝐵X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1630  wcel 2137  {cab 2744  wral 3048   Fn wfn 6042  cfv 6047  Xcixp 8072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-fn 6050  df-ixp 8073
This theorem is referenced by:  ixpeq1d  8084  finixpnum  33705  ioorrnopn  41026  ioorrnopnxr  41028  ovnval  41259  hoicvr  41266  hoidmv1le  41312  hoidmvle  41318  ovnhoi  41321  hspval  41327  ovnlecvr2  41328  hoiqssbl  41343  vonhoire  41390  iunhoiioo  41394  vonioo  41400  vonicc  41403
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