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Theorem elfi 7651
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 7650 . . 3  |-  ( B  e.  W  ->  ( fi `  B )  =  { y  |  E. x  e.  ( ~P B  i^i  Fin ) y  =  |^| x }
)
21eleq2d 2500 . 2  |-  ( B  e.  W  ->  ( A  e.  ( fi `  B )  <->  A  e.  { y  |  E. x  e.  ( ~P B  i^i  Fin ) y  =  |^| x } ) )
3 eqeq1 2439 . . . 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 3096 . 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 693 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 1362    e. wcel 1755   {cab 2419   E.wrex 2706    i^i cin 3315   ~Pcpw 3848   |^|cint 4116   ` cfv 5406   Fincfn 7298   ficfi 7648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-int 4117  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-iota 5369  df-fun 5408  df-fv 5414  df-fi 7649
This theorem is referenced by:  elfi2  7652  elfir  7653  inelfi  7656  fiin  7660  dffi2  7661  elfiun  7668  subbascn  18699  cmpfi  18852  fbasfip  19282  alexsubALTlem4  19463  heibor1lem  28549  elrfi  28872
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