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Theorem elfi2 7422
Description: The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
elfi2  |-  ( B  e.  V  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  ( ( ~P B  i^i  Fin )  \  { (/)
} ) A  = 
|^| x ) )
Distinct variable groups:    x, A    x, B    x, V

Proof of Theorem elfi2
StepHypRef Expression
1 elex 2966 . . 3  |-  ( A  e.  ( fi `  B )  ->  A  e.  _V )
21a1i 11 . 2  |-  ( B  e.  V  ->  ( A  e.  ( fi `  B )  ->  A  e.  _V ) )
3 simpr 449 . . . . 5  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  A  =  |^| x )
4 eldifsni 3930 . . . . . . 7  |-  ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  ->  x  =/=  (/) )
54adantr 453 . . . . . 6  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  x  =/=  (/) )
6 intex 4359 . . . . . 6  |-  ( x  =/=  (/)  <->  |^| x  e.  _V )
75, 6sylib 190 . . . . 5  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  |^| x  e.  _V )
83, 7eqeltrd 2512 . . . 4  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  A  e.  _V )
98rexlimiva 2827 . . 3  |-  ( E. x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } ) A  =  |^| x  ->  A  e.  _V )
109a1i 11 . 2  |-  ( B  e.  V  ->  ( E. x  e.  (
( ~P B  i^i  Fin )  \  { (/) } ) A  =  |^| x  ->  A  e.  _V ) )
11 elfi 7421 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x ) )
12 vprc 4344 . . . . . . . . . . 11  |-  -.  _V  e.  _V
13 elsni 3840 . . . . . . . . . . . . . 14  |-  ( x  e.  { (/) }  ->  x  =  (/) )
1413inteqd 4057 . . . . . . . . . . . . 13  |-  ( x  e.  { (/) }  ->  |^| x  =  |^| (/) )
15 int0 4066 . . . . . . . . . . . . 13  |-  |^| (/)  =  _V
1614, 15syl6eq 2486 . . . . . . . . . . . 12  |-  ( x  e.  { (/) }  ->  |^| x  =  _V )
1716eleq1d 2504 . . . . . . . . . . 11  |-  ( x  e.  { (/) }  ->  (
|^| x  e.  _V  <->  _V  e.  _V ) )
1812, 17mtbiri 296 . . . . . . . . . 10  |-  ( x  e.  { (/) }  ->  -. 
|^| x  e.  _V )
19 simpr 449 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  A  =  |^| x )  ->  A  =  |^| x )
20 simpll 732 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  A  =  |^| x )  ->  A  e.  _V )
2119, 20eqeltrrd 2513 . . . . . . . . . 10  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  A  =  |^| x )  ->  |^| x  e.  _V )
2218, 21nsyl3 114 . . . . . . . . 9  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  A  =  |^| x )  ->  -.  x  e.  { (/) } )
2322biantrud 495 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  A  =  |^| x )  ->  (
x  e.  ( ~P B  i^i  Fin )  <->  ( x  e.  ( ~P B  i^i  Fin )  /\  -.  x  e.  { (/)
} ) ) )
24 eldif 3332 . . . . . . . 8  |-  ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  <->  ( x  e.  ( ~P B  i^i  Fin )  /\  -.  x  e.  { (/) } ) )
2523, 24syl6bbr 256 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  A  =  |^| x )  ->  (
x  e.  ( ~P B  i^i  Fin )  <->  x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } ) ) )
2625pm5.32da 624 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( ( A  = 
|^| x  /\  x  e.  ( ~P B  i^i  Fin ) )  <->  ( A  =  |^| x  /\  x  e.  ( ( ~P B  i^i  Fin )  \  { (/)
} ) ) ) )
27 ancom 439 . . . . . 6  |-  ( ( x  e.  ( ~P B  i^i  Fin )  /\  A  =  |^| x )  <->  ( A  =  |^| x  /\  x  e.  ( ~P B  i^i  Fin ) ) )
28 ancom 439 . . . . . 6  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  <->  ( A  =  |^| x  /\  x  e.  ( ( ~P B  i^i  Fin )  \  { (/)
} ) ) )
2926, 27, 283bitr4g 281 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( ( x  e.  ( ~P B  i^i  Fin )  /\  A  = 
|^| x )  <->  ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/)
} )  /\  A  =  |^| x ) ) )
3029rexbidv2 2730 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x 
<->  E. x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } ) A  =  |^| x ) )
3111, 30bitrd 246 . . 3  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  (
( ~P B  i^i  Fin )  \  { (/) } ) A  =  |^| x ) )
3231expcom 426 . 2  |-  ( B  e.  V  ->  ( A  e.  _V  ->  ( A  e.  ( fi
`  B )  <->  E. x  e.  ( ( ~P B  i^i  Fin )  \  { (/)
} ) A  = 
|^| x ) ) )
332, 10, 32pm5.21ndd 345 1  |-  ( B  e.  V  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  ( ( ~P B  i^i  Fin )  \  { (/)
} ) A  = 
|^| x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   _Vcvv 2958    \ cdif 3319    i^i cin 3321   (/)c0 3630   ~Pcpw 3801   {csn 3816   |^|cint 4052   ` cfv 5457   Fincfn 7112   ficfi 7418
This theorem is referenced by:  fifo  7440  firest  13665  alexsublem  18080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-fi 7419
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