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Theorem ixp0x 8966
Description: An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
Assertion
Ref Expression
ixp0x X𝑥 ∈ ∅ 𝐴 = {∅}

Proof of Theorem ixp0x
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 dfixp 8939 . 2 X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴)}
2 velsn 4642 . . . 4 (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
3 fn0 6699 . . . 4 (𝑓 Fn ∅ ↔ 𝑓 = ∅)
4 ral0 4513 . . . . 5 𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴
54biantru 529 . . . 4 (𝑓 Fn ∅ ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴))
62, 3, 53bitr2i 299 . . 3 (𝑓 ∈ {∅} ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴))
76eqabi 2877 . 2 {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴)}
81, 7eqtr4i 2768 1 X𝑥 ∈ ∅ 𝐴 = {∅}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  c0 4333  {csn 4626   Fn wfn 6556  cfv 6561  Xcixp 8937
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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-fun 6563  df-fn 6564  df-ixp 8938
This theorem is referenced by:  0elixp  8969  ptcmpfi  23821  finixpnum  37612  ioorrnopn  46320  ioorrnopnxr  46322  hoicvr  46563  ovnhoi  46618  ovnlecvr2  46625  hoiqssbl  46640  hoimbl  46646  iunhoiioo  46691
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