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Theorem ixp0x 6844
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 6819 . 2  |-  X_ x  e.  (/)  A  =  {
f  |  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `  x )  e.  A
) }
2 elsn 3655 . . . 4  |-  ( f  e.  { (/) }  <->  f  =  (/) )
3 fn0 5363 . . . 4  |-  ( f  Fn  (/)  <->  f  =  (/) )
4 ral0 3558 . . . . 5  |-  A. x  e.  (/)  ( f `  x )  e.  A
54biantru 491 . . . 4  |-  ( f  Fn  (/)  <->  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `  x
)  e.  A ) )
62, 3, 53bitr2i 264 . . 3  |-  ( f  e.  { (/) }  <->  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `  x )  e.  A
) )
76abbi2i 2394 . 2  |-  { (/) }  =  { f  |  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `
 x )  e.  A ) }
81, 7eqtr4i 2306 1  |-  X_ x  e.  (/)  A  =  { (/)
}
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   (/)c0 3455   {csn 3640    Fn wfn 5250   ` cfv 5255   X_cixp 6817
This theorem is referenced by:  0elixp  6847  ptcmpfi  17504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258  df-ixp 6818
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