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Theorem ixp0x 7402
Description: An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)
Assertion
Ref Expression
ixp0x  |-  X_ x  e.  (/)  A  =  { (/)
}

Proof of Theorem ixp0x
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 dfixp 7376 . 2  |-  X_ x  e.  (/)  A  =  {
f  |  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `  x )  e.  A
) }
2 elsn 4000 . . . 4  |-  ( f  e.  { (/) }  <->  f  =  (/) )
3 fn0 5639 . . . 4  |-  ( f  Fn  (/)  <->  f  =  (/) )
4 ral0 3893 . . . . 5  |-  A. x  e.  (/)  ( f `  x )  e.  A
54biantru 505 . . . 4  |-  ( f  Fn  (/)  <->  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `  x
)  e.  A ) )
62, 3, 53bitr2i 273 . . 3  |-  ( f  e.  { (/) }  <->  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `  x )  e.  A
) )
76abbi2i 2587 . 2  |-  { (/) }  =  { f  |  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `
 x )  e.  A ) }
81, 7eqtr4i 2486 1  |-  X_ x  e.  (/)  A  =  { (/)
}
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2439   A.wral 2799   (/)c0 3746   {csn 3986    Fn wfn 5522   ` cfv 5527   X_cixp 7374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-fun 5529  df-fn 5530  df-ixp 7375
This theorem is referenced by:  0elixp  7405  ptcmpfi  19519  finixpnum  28563
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