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Theorem elfir 7422
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 958 . . . . . 6  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  C_  B )
2 elpw2g 4365 . . . . . 6  |-  ( B  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
31, 2syl5ibr 214 . . . . 5  |-  ( B  e.  V  ->  (
( A  C_  B  /\  A  =/=  (/)  /\  A  e.  Fin )  ->  A  e.  ~P B ) )
43imp 420 . . . 4  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  ~P B
)
5 simpr3 966 . . . 4  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  Fin )
6 elin 3532 . . . 4  |-  ( A  e.  ( ~P B  i^i  Fin )  <->  ( A  e.  ~P B  /\  A  e.  Fin ) )
74, 5, 6sylanbrc 647 . . 3  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  ( ~P B  i^i  Fin ) )
8 eqid 2438 . . 3  |-  |^| A  =  |^| A
9 inteq 4055 . . . . 5  |-  ( x  =  A  ->  |^| x  =  |^| A )
109eqeq2d 2449 . . . 4  |-  ( x  =  A  ->  ( |^| A  =  |^| x  <->  |^| A  =  |^| A
) )
1110rspcev 3054 . . 3  |-  ( ( A  e.  ( ~P B  i^i  Fin )  /\  |^| A  =  |^| A )  ->  E. x  e.  ( ~P B  i^i  Fin ) |^| A  = 
|^| x )
127, 8, 11sylancl 645 . 2  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  E. x  e.  ( ~P B  i^i  Fin ) |^| A  =  |^| x
)
13 simp2 959 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  =/=  (/) )
14 intex 4358 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
1513, 14sylib 190 . . 3  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  |^| A  e.  _V )
16 id 21 . . 3  |-  ( B  e.  V  ->  B  e.  V )
17 elfi 7420 . . 3  |-  ( (
|^| A  e.  _V  /\  B  e.  V )  ->  ( |^| A  e.  ( fi `  B
)  <->  E. x  e.  ( ~P B  i^i  Fin ) |^| A  =  |^| x ) )
1815, 16, 17syl2anr 466 . 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
) )
1912, 18mpbird 225 1  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  e.  ( fi
`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   _Vcvv 2958    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   |^|cint 4052   ` cfv 5456   Fincfn 7111   ficfi 7417
This theorem is referenced by:  intrnfi  7423  ssfii  7426  elfiun  7437  ptbasfi  17615  fbssint  17872  filintn0  17895  alexsublem  18077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-fi 7418
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