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

Theorem elfi 7167
Description: Specific properties of an element of  ( fi `  B ). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
Assertion
Ref Expression
elfi  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x ) )
Distinct variable groups:    x, A    x, B    x, V    x, W

Proof of Theorem elfi
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fival 7166 . . 3  |-  ( B  e.  W  ->  ( fi `  B )  =  { y  |  E. x  e.  ( ~P B  i^i  Fin ) y  =  |^| x }
)
21eleq2d 2350 . 2  |-  ( B  e.  W  ->  ( A  e.  ( fi `  B )  <->  A  e.  { y  |  E. x  e.  ( ~P B  i^i  Fin ) y  =  |^| x } ) )
3 eqeq1 2289 . . . 4  |-  ( y  =  A  ->  (
y  =  |^| x  <->  A  =  |^| x ) )
43rexbidv 2564 . . 3  |-  ( y  =  A  ->  ( E. x  e.  ( ~P B  i^i  Fin )
y  =  |^| x  <->  E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x ) )
54elabg 2915 . 2  |-  ( A  e.  V  ->  ( A  e.  { y  |  E. x  e.  ( ~P B  i^i  Fin ) y  =  |^| x }  <->  E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x ) )
62, 5sylan9bbr 681 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    i^i cin 3151   ~Pcpw 3625   |^|cint 3862   ` cfv 5255   Fincfn 6863   ficfi 7164
This theorem is referenced by:  elfi2  7168  elfir  7169  fiin  7175  dffi2  7176  elfiun  7183  subbascn  16984  cmpfi  17135  fbasfip  17563  alexsubALTlem4  17744  heibor1lem  26533  elrfi  26769
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-fi 7165
  Copyright terms: Public domain W3C validator