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Theorem ixp0x 8863
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 8836 . 2 X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴)}
2 velsn 4602 . . . 4 (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
3 fn0 6632 . . . 4 (𝑓 Fn ∅ ↔ 𝑓 = ∅)
4 ral0 4470 . . . . 5 𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴
54biantru 530 . . . 4 (𝑓 Fn ∅ ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴))
62, 3, 53bitr2i 298 . . 3 (𝑓 ∈ {∅} ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴))
76abbi2i 2873 . 2 {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴)}
81, 7eqtr4i 2767 1 X𝑥 ∈ ∅ 𝐴 = {∅}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  {cab 2713  wral 3064  c0 4282  {csn 4586   Fn wfn 6491  cfv 6496  Xcixp 8834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-fun 6498  df-fn 6499  df-ixp 8835
This theorem is referenced by:  0elixp  8866  ptcmpfi  23162  finixpnum  36053  ioorrnopn  44517  ioorrnopnxr  44519  hoicvr  44760  ovnhoi  44815  ovnlecvr2  44822  hoiqssbl  44837  hoimbl  44843  iunhoiioo  44888
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