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Theorem elfi2 7664
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 2981 . . 3  |-  ( A  e.  ( fi `  B )  ->  A  e.  _V )
21a1i 11 . 2  |-  ( B  e.  V  ->  ( A  e.  ( fi `  B )  ->  A  e.  _V ) )
3 simpr 461 . . . . 5  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  A  =  |^| x )
4 eldifsni 4001 . . . . . . 7  |-  ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  ->  x  =/=  (/) )
54adantr 465 . . . . . 6  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  x  =/=  (/) )
6 intex 4448 . . . . . 6  |-  ( x  =/=  (/)  <->  |^| x  e.  _V )
75, 6sylib 196 . . . . 5  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  |^| x  e.  _V )
83, 7eqeltrd 2517 . . . 4  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  A  e.  _V )
98rexlimiva 2836 . . 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 7663 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x ) )
12 vprc 4430 . . . . . . . . . . 11  |-  -.  _V  e.  _V
13 elsni 3902 . . . . . . . . . . . . . 14  |-  ( x  e.  { (/) }  ->  x  =  (/) )
1413inteqd 4133 . . . . . . . . . . . . 13  |-  ( x  e.  { (/) }  ->  |^| x  =  |^| (/) )
15 int0 4142 . . . . . . . . . . . . 13  |-  |^| (/)  =  _V
1614, 15syl6eq 2491 . . . . . . . . . . . 12  |-  ( x  e.  { (/) }  ->  |^| x  =  _V )
1716eleq1d 2509 . . . . . . . . . . 11  |-  ( x  e.  { (/) }  ->  (
|^| x  e.  _V  <->  _V  e.  _V ) )
1812, 17mtbiri 303 . . . . . . . . . 10  |-  ( x  e.  { (/) }  ->  -. 
|^| x  e.  _V )
19 simpr 461 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  A  =  |^| x )  ->  A  =  |^| x )
20 simpll 753 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  A  =  |^| x )  ->  A  e.  _V )
2119, 20eqeltrrd 2518 . . . . . . . . . 10  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  A  =  |^| x )  ->  |^| x  e.  _V )
2218, 21nsyl3 119 . . . . . . . . 9  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  A  =  |^| x )  ->  -.  x  e.  { (/) } )
2322biantrud 507 . . . . . . . 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 3338 . . . . . . . 8  |-  ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  <->  ( x  e.  ( ~P B  i^i  Fin )  /\  -.  x  e.  { (/) } ) )
2523, 24syl6bbr 263 . . . . . . 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 641 . . . . . 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 450 . . . . . 6  |-  ( ( x  e.  ( ~P B  i^i  Fin )  /\  A  =  |^| x )  <->  ( A  =  |^| x  /\  x  e.  ( ~P B  i^i  Fin ) ) )
28 ancom 450 . . . . . 6  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  <->  ( A  =  |^| x  /\  x  e.  ( ( ~P B  i^i  Fin )  \  { (/)
} ) ) )
2926, 27, 283bitr4g 288 . . . . 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 2738 . . . 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 253 . . 3  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  (
( ~P B  i^i  Fin )  \  { (/) } ) A  =  |^| x ) )
3231expcom 435 . 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 354 1  |-  ( B  e.  V  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  ( ( ~P B  i^i  Fin )  \  { (/)
} ) A  = 
|^| x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716   _Vcvv 2972    \ cdif 3325    i^i cin 3327   (/)c0 3637   ~Pcpw 3860   {csn 3877   |^|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:  fifo  7682  firest  14371  alexsublem  19616
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