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

Theorem elfi 7885
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 7884 . . 3  |-  ( B  e.  W  ->  ( fi `  B )  =  { y  |  E. x  e.  ( ~P B  i^i  Fin ) y  =  |^| x }
)
21eleq2d 2537 . 2  |-  ( B  e.  W  ->  ( A  e.  ( fi `  B )  <->  A  e.  { y  |  E. x  e.  ( ~P B  i^i  Fin ) y  =  |^| x } ) )
3 eqeq1 2471 . . . 4  |-  ( y  =  A  ->  (
y  =  |^| x  <->  A  =  |^| x ) )
43rexbidv 2978 . . 3  |-  ( y  =  A  ->  ( E. x  e.  ( ~P B  i^i  Fin )
y  =  |^| x  <->  E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x ) )
54elabg 3256 . 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 700 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2818    i^i cin 3480   ~Pcpw 4016   |^|cint 4288   ` cfv 5594   Fincfn 7528   ficfi 7882
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-fi 7883
This theorem is referenced by:  elfi2  7886  elfir  7887  inelfi  7890  fiin  7894  dffi2  7895  elfiun  7902  subbascn  19623  cmpfi  19776  fbasfip  20237  alexsubALTlem4  20418  heibor1lem  30232  elrfi  30554
  Copyright terms: Public domain W3C validator