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Theorem ixp0x 7550
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 7524 . 2  |-  X_ x  e.  (/)  A  =  {
f  |  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `  x )  e.  A
) }
2 elsn 3982 . . . 4  |-  ( f  e.  { (/) }  <->  f  =  (/) )
3 fn0 5695 . . . 4  |-  ( f  Fn  (/)  <->  f  =  (/) )
4 ral0 3874 . . . . 5  |-  A. x  e.  (/)  ( f `  x )  e.  A
54biantru 508 . . . 4  |-  ( f  Fn  (/)  <->  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `  x
)  e.  A ) )
62, 3, 53bitr2i 277 . . 3  |-  ( f  e.  { (/) }  <->  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `  x )  e.  A
) )
76abbi2i 2566 . 2  |-  { (/) }  =  { f  |  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `
 x )  e.  A ) }
81, 7eqtr4i 2476 1  |-  X_ x  e.  (/)  A  =  { (/)
}
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371    = wceq 1444    e. wcel 1887   {cab 2437   A.wral 2737   (/)c0 3731   {csn 3968    Fn wfn 5577   ` cfv 5582   X_cixp 7522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-fun 5584  df-fn 5585  df-ixp 7523
This theorem is referenced by:  0elixp  7553  ptcmpfi  20828  finixpnum  31930  hoicvr  38370  ovnhoi  38425  ovnlecvr2  38432  hoiqssbl  38447  hoimbl  38453
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