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Theorem ixp0x 6860
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 6835 . 2  |-  X_ x  e.  (/)  A  =  {
f  |  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `  x )  e.  A
) }
2 elsn 3668 . . . 4  |-  ( f  e.  { (/) }  <->  f  =  (/) )
3 fn0 5379 . . . 4  |-  ( f  Fn  (/)  <->  f  =  (/) )
4 ral0 3571 . . . . 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 2407 . 2  |-  { (/) }  =  { f  |  ( f  Fn  (/)  /\  A. x  e.  (/)  ( f `
 x )  e.  A ) }
81, 7eqtr4i 2319 1  |-  X_ x  e.  (/)  A  =  { (/)
}
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   (/)c0 3468   {csn 3653    Fn wfn 5266   ` cfv 5271   X_cixp 6833
This theorem is referenced by:  0elixp  6863  ptcmpfi  17520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274  df-ixp 6834
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