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Theorem elfir 7665
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 988 . . . . . 6  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  C_  B )
2 elpw2g 4455 . . . . . 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 996 . . . 4  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  Fin )
64, 5elind 3540 . . 3  |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  A  e.  ( ~P B  i^i  Fin ) )
7 eqid 2443 . . 3  |-  |^| A  =  |^| A
8 inteq 4131 . . . . 5  |-  ( x  =  A  ->  |^| x  =  |^| A )
98eqeq2d 2454 . . . 4  |-  ( x  =  A  ->  ( |^| A  =  |^| x  <->  |^| A  =  |^| A
) )
109rspcev 3073 . . 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 989 . . . 4  |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  A  =/=  (/) )
13 intex 4448 . . . 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 7663 . . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716   _Vcvv 2972    i^i cin 3327    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   |^|cint 4128   ` cfv 5418   Fincfn 7310   ficfi 7660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-fi 7661
This theorem is referenced by:  intrnfi  7666  ssfii  7669  elfiun  7680  ptbasfi  19154  fbssint  19411  filintn0  19434  alexsublem  19616
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