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Theorem elfi 7953
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 7952 . . 3  |-  ( B  e.  W  ->  ( fi `  B )  =  { y  |  E. x  e.  ( ~P B  i^i  Fin ) y  =  |^| x }
)
21eleq2d 2525 . 2  |-  ( B  e.  W  ->  ( A  e.  ( fi `  B )  <->  A  e.  { y  |  E. x  e.  ( ~P B  i^i  Fin ) y  =  |^| x } ) )
3 eqeq1 2466 . . . 4  |-  ( y  =  A  ->  (
y  =  |^| x  <->  A  =  |^| x ) )
43rexbidv 2913 . . 3  |-  ( y  =  A  ->  ( E. x  e.  ( ~P B  i^i  Fin )
y  =  |^| x  <->  E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x ) )
54elabg 3198 . 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 712 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 189    /\ wa 375    = wceq 1455    e. wcel 1898   {cab 2448   E.wrex 2750    i^i cin 3415   ~Pcpw 3963   |^|cint 4248   ` cfv 5601   Fincfn 7595   ficfi 7950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-int 4249  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-fi 7951
This theorem is referenced by:  elfi2  7954  elfir  7955  inelfi  7958  fiin  7962  dffi2  7963  elfiun  7970  subbascn  20319  cmpfi  20472  fbasfip  20932  alexsubALTlem4  21114  heibor1lem  32186  elrfi  35581
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