MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ixp0x Structured version   Visualization version   GIF version

Theorem ixp0x 8965
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 8938 . 2 X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴)}
2 velsn 4647 . . . 4 (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
3 fn0 6700 . . . 4 (𝑓 Fn ∅ ↔ 𝑓 = ∅)
4 ral0 4519 . . . . 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 2875 . 2 {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴)}
81, 7eqtr4i 2766 1 X𝑥 ∈ ∅ 𝐴 = {∅}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  c0 4339  {csn 4631   Fn wfn 6558  cfv 6563  Xcixp 8936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-fun 6565  df-fn 6566  df-ixp 8937
This theorem is referenced by:  0elixp  8968  ptcmpfi  23837  finixpnum  37592  ioorrnopn  46261  ioorrnopnxr  46263  hoicvr  46504  ovnhoi  46559  ovnlecvr2  46566  hoiqssbl  46581  hoimbl  46587  iunhoiioo  46632
  Copyright terms: Public domain W3C validator