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Theorem elfi2 7863
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 3115 . . 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 4146 . . . . . . 7  |-  ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  ->  x  =/=  (/) )
54adantr 465 . . . . . 6  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  x  =/=  (/) )
6 intex 4596 . . . . . 6  |-  ( x  =/=  (/)  <->  |^| x  e.  _V )
75, 6sylib 196 . . . . 5  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  |^| x  e.  _V )
83, 7eqeltrd 2548 . . . 4  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  A  e.  _V )
98rexlimiva 2944 . . 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 7862 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x ) )
12 vprc 4578 . . . . . . . . . . 11  |-  -.  _V  e.  _V
13 elsni 4045 . . . . . . . . . . . . . 14  |-  ( x  e.  { (/) }  ->  x  =  (/) )
1413inteqd 4280 . . . . . . . . . . . . 13  |-  ( x  e.  { (/) }  ->  |^| x  =  |^| (/) )
15 int0 4289 . . . . . . . . . . . . 13  |-  |^| (/)  =  _V
1614, 15syl6eq 2517 . . . . . . . . . . . 12  |-  ( x  e.  { (/) }  ->  |^| x  =  _V )
1716eleq1d 2529 . . . . . . . . . . 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 2549 . . . . . . . . . 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 3479 . . . . . . . 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 2962 . . . 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 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808   _Vcvv 3106    \ cdif 3466    i^i cin 3468   (/)c0 3778   ~Pcpw 4003   {csn 4020   |^|cint 4275   ` cfv 5579   Fincfn 7506   ficfi 7859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-int 4276  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-fi 7860
This theorem is referenced by:  fifo  7881  firest  14677  alexsublem  20272
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