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Theorem elfi2 7258
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 2872 . . 3  |-  ( A  e.  ( fi `  B )  ->  A  e.  _V )
21a1i 10 . 2  |-  ( B  e.  V  ->  ( A  e.  ( fi `  B )  ->  A  e.  _V ) )
3 simpr 447 . . . . 5  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  A  =  |^| x )
4 eldifsni 3826 . . . . . . 7  |-  ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  ->  x  =/=  (/) )
54adantr 451 . . . . . 6  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  x  =/=  (/) )
6 intex 4248 . . . . . 6  |-  ( x  =/=  (/)  <->  |^| x  e.  _V )
75, 6sylib 188 . . . . 5  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  |^| x  e.  _V )
83, 7eqeltrd 2432 . . . 4  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  ->  A  e.  _V )
98rexlimiva 2738 . . 3  |-  ( E. x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } ) A  =  |^| x  ->  A  e.  _V )
109a1i 10 . 2  |-  ( B  e.  V  ->  ( E. x  e.  (
( ~P B  i^i  Fin )  \  { (/) } ) A  =  |^| x  ->  A  e.  _V ) )
11 elfi 7257 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x ) )
12 vprc 4233 . . . . . . . . . . 11  |-  -.  _V  e.  _V
13 elsni 3740 . . . . . . . . . . . . . 14  |-  ( x  e.  { (/) }  ->  x  =  (/) )
1413inteqd 3948 . . . . . . . . . . . . 13  |-  ( x  e.  { (/) }  ->  |^| x  =  |^| (/) )
15 int0 3957 . . . . . . . . . . . . 13  |-  |^| (/)  =  _V
1614, 15syl6eq 2406 . . . . . . . . . . . 12  |-  ( x  e.  { (/) }  ->  |^| x  =  _V )
1716eleq1d 2424 . . . . . . . . . . 11  |-  ( x  e.  { (/) }  ->  (
|^| x  e.  _V  <->  _V  e.  _V ) )
1812, 17mtbiri 294 . . . . . . . . . 10  |-  ( x  e.  { (/) }  ->  -. 
|^| x  e.  _V )
19 simpr 447 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  A  =  |^| x )  ->  A  =  |^| x )
20 simpll 730 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  A  =  |^| x )  ->  A  e.  _V )
2119, 20eqeltrrd 2433 . . . . . . . . . 10  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  A  =  |^| x )  ->  |^| x  e.  _V )
2218, 21nsyl3 111 . . . . . . . . 9  |-  ( ( ( A  e.  _V  /\  B  e.  V )  /\  A  =  |^| x )  ->  -.  x  e.  { (/) } )
2322biantrud 493 . . . . . . . 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 3238 . . . . . . . 8  |-  ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  <->  ( x  e.  ( ~P B  i^i  Fin )  /\  -.  x  e.  { (/) } ) )
2523, 24syl6bbr 254 . . . . . . 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 622 . . . . . 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 437 . . . . . 6  |-  ( ( x  e.  ( ~P B  i^i  Fin )  /\  A  =  |^| x )  <->  ( A  =  |^| x  /\  x  e.  ( ~P B  i^i  Fin ) ) )
28 ancom 437 . . . . . 6  |-  ( ( x  e.  ( ( ~P B  i^i  Fin )  \  { (/) } )  /\  A  =  |^| x )  <->  ( A  =  |^| x  /\  x  e.  ( ( ~P B  i^i  Fin )  \  { (/)
} ) ) )
2926, 27, 283bitr4g 279 . . . . 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 2642 . . . 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 244 . . 3  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  (
( ~P B  i^i  Fin )  \  { (/) } ) A  =  |^| x ) )
3231expcom 424 . 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 343 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 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620   _Vcvv 2864    \ cdif 3225    i^i cin 3227   (/)c0 3531   ~Pcpw 3701   {csn 3716   |^|cint 3943   ` cfv 5337   Fincfn 6951   ficfi 7254
This theorem is referenced by:  fifo  7275  firest  13436  alexsublem  17840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-int 3944  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-fi 7255
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