MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elixp Structured version   Unicode version

Theorem elixp 7473
Description: Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)
Hypothesis
Ref Expression
elixp.1  |-  F  e. 
_V
Assertion
Ref Expression
elixp  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
Distinct variable groups:    x, F    x, A
Allowed substitution hint:    B( x)

Proof of Theorem elixp
StepHypRef Expression
1 elixp2 7470 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
2 elixp.1 . . 3  |-  F  e. 
_V
3 3anass 977 . . 3  |-  ( ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  <->  ( F  e.  _V  /\  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) ) )
42, 3mpbiran 916 . 2  |-  ( ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) )
51, 4bitri 249 1  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767   A.wral 2814   _Vcvv 3113    Fn wfn 5581   ` cfv 5586   X_cixp 7466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-ixp 7467
This theorem is referenced by:  elixpconst  7474  ixpin  7491  ixpiin  7492  resixpfo  7504  elixpsn  7505  boxriin  7508  boxcutc  7509  ixpfi2  7814  ixpiunwdom  8013  dfac9  8512  ac9  8859  ac9s  8869  konigthlem  8939  xpscf  14817  cofucl  15111  yonedalem3  15403  psrbaglefi  17794  psrbaglefiOLD  17795  ptpjpre1  19807  ptpjcn  19847  ptpjopn  19848  ptclsg  19851  dfac14  19854  pthaus  19874  xkopt  19891  ptcmplem2  20288  ptcmplem3  20289  ptcmplem4  20290  prdsbl  20729  prdsxmslem2  20767  eulerpartlemb  27947  ptpcon  28318  finixpnum  29615  ptrest  29625  inixp  29822  prdstotbnd  29893
  Copyright terms: Public domain W3C validator