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Theorem elfir 7169
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 955 . . . . . 6  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  C_  B )
2 elpw2g 4174 . . . . . 6  |-  ( B  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
31, 2syl5ibr 212 . . . . 5  |-  ( B  e.  V  ->  (
( A  C_  B  /\  A  =/=  (/)  /\  A  e.  Fin )  ->  A  e.  ~P B ) )
43imp 418 . . . 4  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  ~P B
)
5 simpr3 963 . . . 4  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  Fin )
6 elin 3358 . . . 4  |-  ( A  e.  ( ~P B  i^i  Fin )  <->  ( A  e.  ~P B  /\  A  e.  Fin ) )
74, 5, 6sylanbrc 645 . . 3  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  ( ~P B  i^i  Fin ) )
8 eqid 2283 . . 3  |-  |^| A  =  |^| A
9 inteq 3865 . . . . 5  |-  ( x  =  A  ->  |^| x  =  |^| A )
109eqeq2d 2294 . . . 4  |-  ( x  =  A  ->  ( |^| A  =  |^| x  <->  |^| A  =  |^| A
) )
1110rspcev 2884 . . 3  |-  ( ( A  e.  ( ~P B  i^i  Fin )  /\  |^| A  =  |^| A )  ->  E. x  e.  ( ~P B  i^i  Fin ) |^| A  = 
|^| x )
127, 8, 11sylancl 643 . 2  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  E. x  e.  ( ~P B  i^i  Fin ) |^| A  =  |^| x
)
13 simp2 956 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  =/=  (/) )
14 intex 4167 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
1513, 14sylib 188 . . 3  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  |^| A  e.  _V )
16 id 19 . . 3  |-  ( B  e.  V  ->  B  e.  V )
17 elfi 7167 . . 3  |-  ( (
|^| A  e.  _V  /\  B  e.  V )  ->  ( |^| A  e.  ( fi `  B
)  <->  E. x  e.  ( ~P B  i^i  Fin ) |^| A  =  |^| x ) )
1815, 16, 17syl2anr 464 . 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 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   |^|cint 3862   ` cfv 5255   Fincfn 6863   ficfi 7164
This theorem is referenced by:  intrnfi  7170  ssfii  7172  elfiun  7183  ptbasfi  17276  fbssint  17533  filintn0  17556  alexsublem  17738  filint2  25553
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-fi 7165
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