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

Theorem ixp0x 8143
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 8117 . 2 X𝑥 ∈ ∅ 𝐴 = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴)}
2 velsn 4352 . . . 4 (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
3 fn0 6191 . . . 4 (𝑓 Fn ∅ ↔ 𝑓 = ∅)
4 ral0 4237 . . . . 5 𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴
54biantru 525 . . . 4 (𝑓 Fn ∅ ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴))
62, 3, 53bitr2i 290 . . 3 (𝑓 ∈ {∅} ↔ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴))
76abbi2i 2881 . 2 {∅} = {𝑓 ∣ (𝑓 Fn ∅ ∧ ∀𝑥 ∈ ∅ (𝑓𝑥) ∈ 𝐴)}
81, 7eqtr4i 2790 1 X𝑥 ∈ ∅ 𝐴 = {∅}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  wcel 2155  {cab 2751  wral 3055  c0 4081  {csn 4336   Fn wfn 6065  cfv 6070  Xcixp 8115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-fun 6072  df-fn 6073  df-ixp 8116
This theorem is referenced by:  0elixp  8146  ptcmpfi  21899  finixpnum  33821  ioorrnopn  41165  ioorrnopnxr  41167  hoicvr  41405  ovnhoi  41460  ovnlecvr2  41467  hoiqssbl  41482  hoimbl  41488  iunhoiioo  41533
  Copyright terms: Public domain W3C validator