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Theorem ixp0x 6628
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 6602 . 2  |-  X_ x  e.  (/)  A  =  {
f  |  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `  x )  e.  A
) }
2 velsn 3549 . . . 4  |-  ( f  e.  { (/) }  <->  f  =  (/) )
3 fn0 5250 . . . 4  |-  ( f  Fn  (/)  <->  f  =  (/) )
4 ral0 3469 . . . . 5  |-  A. x  e.  (/)  ( f `  x )  e.  A
54biantru 300 . . . 4  |-  ( f  Fn  (/)  <->  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `  x
)  e.  A ) )
62, 3, 53bitr2i 207 . . 3  |-  ( f  e.  { (/) }  <->  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `  x )  e.  A
) )
76abbi2i 2255 . 2  |-  { (/) }  =  { f  |  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `
 x )  e.  A ) }
81, 7eqtr4i 2164 1  |-  X_ x  e.  (/)  A  =  { (/)
}
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1332    e. wcel 1481   {cab 2126   A.wral 2417   (/)c0 3368   {csn 3532    Fn wfn 5126   ` cfv 5131   X_cixp 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-fun 5133  df-fn 5134  df-ixp 6601
This theorem is referenced by:  0elixp  6631
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