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Theorem elfir 7185
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 4190 . . . . . 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 3371 . . . 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 2296 . . 3  |-  |^| A  =  |^| A
9 inteq 3881 . . . . 5  |-  ( x  =  A  ->  |^| x  =  |^| A )
109eqeq2d 2307 . . . 4  |-  ( x  =  A  ->  ( |^| A  =  |^| x  <->  |^| A  =  |^| A
) )
1110rspcev 2897 . . 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 4183 . . . 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 7183 . . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   |^|cint 3878   ` cfv 5271   Fincfn 6879   ficfi 7180
This theorem is referenced by:  intrnfi  7186  ssfii  7188  elfiun  7199  ptbasfi  17292  fbssint  17549  filintn0  17572  alexsublem  17754  filint2  25656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-fi 7181
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