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Theorem elfi 7418
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 7417 . . 3  |-  ( B  e.  W  ->  ( fi `  B )  =  { y  |  E. x  e.  ( ~P B  i^i  Fin ) y  =  |^| x }
)
21eleq2d 2503 . 2  |-  ( B  e.  W  ->  ( A  e.  ( fi `  B )  <->  A  e.  { y  |  E. x  e.  ( ~P B  i^i  Fin ) y  =  |^| x } ) )
3 eqeq1 2442 . . . 4  |-  ( y  =  A  ->  (
y  =  |^| x  <->  A  =  |^| x ) )
43rexbidv 2726 . . 3  |-  ( y  =  A  ->  ( E. x  e.  ( ~P B  i^i  Fin )
y  =  |^| x  <->  E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x ) )
54elabg 3083 . 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 682 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2706    i^i cin 3319   ~Pcpw 3799   |^|cint 4050   ` cfv 5454   Fincfn 7109   ficfi 7415
This theorem is referenced by:  elfi2  7419  elfir  7420  inelfi  7423  fiin  7427  dffi2  7428  elfiun  7435  subbascn  17318  cmpfi  17471  fbasfip  17900  alexsubALTlem4  18081  heibor1lem  26518  elrfi  26748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-fi 7416
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