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Theorem elfir 7887
Description: Sufficient condition for an element of  ( fi `  B ). (Contributed by Mario Carneiro, 24-Nov-2013.)
Assertion
Ref Expression
elfir  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  e.  ( fi
`  B ) )

Proof of Theorem elfir
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp1 996 . . . . . 6  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  C_  B )
2 elpw2g 4616 . . . . . 6  |-  ( B  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
31, 2syl5ibr 221 . . . . 5  |-  ( B  e.  V  ->  (
( A  C_  B  /\  A  =/=  (/)  /\  A  e.  Fin )  ->  A  e.  ~P B ) )
43imp 429 . . . 4  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  ~P B
)
5 simpr3 1004 . . . 4  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  Fin )
64, 5elind 3693 . . 3  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  ( ~P B  i^i  Fin ) )
7 eqid 2467 . . 3  |-  |^| A  =  |^| A
8 inteq 4291 . . . . 5  |-  ( x  =  A  ->  |^| x  =  |^| A )
98eqeq2d 2481 . . . 4  |-  ( x  =  A  ->  ( |^| A  =  |^| x  <->  |^| A  =  |^| A
) )
109rspcev 3219 . . 3  |-  ( ( A  e.  ( ~P B  i^i  Fin )  /\  |^| A  =  |^| A )  ->  E. x  e.  ( ~P B  i^i  Fin ) |^| A  = 
|^| x )
116, 7, 10sylancl 662 . 2  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  E. x  e.  ( ~P B  i^i  Fin ) |^| A  =  |^| x
)
12 simp2 997 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  =/=  (/) )
13 intex 4609 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
1412, 13sylib 196 . . 3  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  |^| A  e.  _V )
15 id 22 . . 3  |-  ( B  e.  V  ->  B  e.  V )
16 elfi 7885 . . 3  |-  ( (
|^| A  e.  _V  /\  B  e.  V )  ->  ( |^| A  e.  ( fi `  B
)  <->  E. x  e.  ( ~P B  i^i  Fin ) |^| A  =  |^| x ) )
1714, 15, 16syl2anr 478 . 2  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  -> 
( |^| A  e.  ( fi `  B )  <->  E. x  e.  ( ~P B  i^i  Fin ) |^| A  =  |^| x
) )
1811, 17mpbird 232 1  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  e.  ( fi
`  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   _Vcvv 3118    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~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:  intrnfi  7888  ssfii  7891  elfiun  7902  ptbasfi  19950  fbssint  20207  filintn0  20230  alexsublem  20412
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