Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cnfilca Unicode version

Theorem cnfilca 25659
Description: Condition to have a filter finer than a given filter and containing a set  A. Bourbaki T.G. I.37 cor. 1 (Contributed by FL, 27-Apr-2008.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
cnfilca  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  ( E. g  e.  ( Fil `  X ) ( A  e.  g  /\  F  C_  g )  <->  A. x  e.  F  ( x  i^i  A )  =/=  (/) ) )
Distinct variable groups:    A, g    g, F    g, X    x, A    x, F    x, X

Proof of Theorem cnfilca
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 filtop 17566 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
2 ssexg 4176 . . . . . . 7  |-  ( ( A  C_  X  /\  X  e.  F )  ->  A  e.  _V )
31, 2sylan2 460 . . . . . 6  |-  ( ( A  C_  X  /\  F  e.  ( Fil `  X ) )  ->  A  e.  _V )
43ancoms 439 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  A  e.  _V )
543adant3 975 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  A  e. 
_V )
6 snssg 3767 . . . . . 6  |-  ( A  e.  _V  ->  ( A  e.  g  <->  { A }  C_  g ) )
76anbi1d 685 . . . . 5  |-  ( A  e.  _V  ->  (
( A  e.  g  /\  F  C_  g
)  <->  ( { A }  C_  g  /\  F  C_  g ) ) )
8 unss 3362 . . . . 5  |-  ( ( { A }  C_  g  /\  F  C_  g
)  <->  ( { A }  u.  F )  C_  g )
97, 8syl6bb 252 . . . 4  |-  ( A  e.  _V  ->  (
( A  e.  g  /\  F  C_  g
)  <->  ( { A }  u.  F )  C_  g ) )
105, 9syl 15 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  ( ( A  e.  g  /\  F  C_  g )  <->  ( { A }  u.  F
)  C_  g )
)
1110rexbidv 2577 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  ( E. g  e.  ( Fil `  X ) ( A  e.  g  /\  F  C_  g )  <->  E. g  e.  ( Fil `  X
) ( { A }  u.  F )  C_  g ) )
12 simplr 731 . . . . . . 7  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  C_  X
)  /\  A  =/=  (/) )  ->  A  C_  X
)
13 simpr 447 . . . . . . 7  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  C_  X
)  /\  A  =/=  (/) )  ->  A  =/=  (/) )
141ad2antrr 706 . . . . . . 7  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  C_  X
)  /\  A  =/=  (/) )  ->  X  e.  F )
15 snfbas 17577 . . . . . . 7  |-  ( ( A  C_  X  /\  A  =/=  (/)  /\  X  e.  F )  ->  { A }  e.  ( fBas `  X ) )
1612, 13, 14, 15syl3anc 1182 . . . . . 6  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  C_  X
)  /\  A  =/=  (/) )  ->  { A }  e.  ( fBas `  X ) )
17 fbsspw 17543 . . . . . 6  |-  ( { A }  e.  (
fBas `  X )  ->  { A }  C_  ~P X )
1816, 17syl 15 . . . . 5  |-  ( ( ( F  e.  ( Fil `  X )  /\  A  C_  X
)  /\  A  =/=  (/) )  ->  { A }  C_  ~P X )
19183impa 1146 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  { A }  C_  ~P X )
20 filsspw 17562 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  F  C_  ~P X )
21203ad2ant1 976 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  F  C_  ~P X )
2219, 21unssd 3364 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  ( { A }  u.  F
)  C_  ~P X
)
2313ad2ant1 976 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  X  e.  F )
24 ssn0 3500 . . . 4  |-  ( ( A  C_  X  /\  A  =/=  (/) )  ->  X  =/=  (/) )
25243adant1 973 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  X  =/=  (/) )
26 efilcp 25655 . . 3  |-  ( ( ( { A }  u.  F )  C_  ~P X  /\  X  e.  F  /\  X  =/=  (/) )  -> 
( -.  (/)  e.  ( fi `  ( { A }  u.  F
) )  <->  E. g  e.  ( Fil `  X
) ( { A }  u.  F )  C_  g ) )
2722, 23, 25, 26syl3anc 1182 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  ( -.  (/)  e.  ( fi `  ( { A }  u.  F ) )  <->  E. g  e.  ( Fil `  X
) ( { A }  u.  F )  C_  g ) )
28163impa 1146 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  { A }  e.  ( fBas `  X ) )
29 filfbas 17559 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
30293ad2ant1 976 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  F  e.  ( fBas `  X
) )
31 fbunfip 17580 . . . 4  |-  ( ( { A }  e.  ( fBas `  X )  /\  F  e.  ( fBas `  X ) )  ->  ( -.  (/)  e.  ( fi `  ( { A }  u.  F
) )  <->  A. y  e.  { A } A. x  e.  F  (
y  i^i  x )  =/=  (/) ) )
3228, 30, 31syl2anc 642 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  ( -.  (/)  e.  ( fi `  ( { A }  u.  F ) )  <->  A. y  e.  { A } A. x  e.  F  (
y  i^i  x )  =/=  (/) ) )
33 ineq1 3376 . . . . . . . 8  |-  ( y  =  A  ->  (
y  i^i  x )  =  ( A  i^i  x ) )
34 incom 3374 . . . . . . . 8  |-  ( A  i^i  x )  =  ( x  i^i  A
)
3533, 34syl6eq 2344 . . . . . . 7  |-  ( y  =  A  ->  (
y  i^i  x )  =  ( x  i^i 
A ) )
3635neeq1d 2472 . . . . . 6  |-  ( y  =  A  ->  (
( y  i^i  x
)  =/=  (/)  <->  ( x  i^i  A )  =/=  (/) ) )
3736ralbidv 2576 . . . . 5  |-  ( y  =  A  ->  ( A. x  e.  F  ( y  i^i  x
)  =/=  (/)  <->  A. x  e.  F  ( x  i^i  A )  =/=  (/) ) )
3837ralsng 3685 . . . 4  |-  ( A  e.  _V  ->  ( A. y  e.  { A } A. x  e.  F  ( y  i^i  x
)  =/=  (/)  <->  A. x  e.  F  ( x  i^i  A )  =/=  (/) ) )
395, 38syl 15 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  ( A. y  e.  { A } A. x  e.  F  ( y  i^i  x
)  =/=  (/)  <->  A. x  e.  F  ( x  i^i  A )  =/=  (/) ) )
4032, 39bitrd 244 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  ( -.  (/)  e.  ( fi `  ( { A }  u.  F ) )  <->  A. x  e.  F  ( x  i^i  A )  =/=  (/) ) )
4111, 27, 403bitr2d 272 1  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  ( E. g  e.  ( Fil `  X ) ( A  e.  g  /\  F  C_  g )  <->  A. x  e.  F  ( x  i^i  A )  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   ` cfv 5271   ficfi 7180   fBascfbas 17534   Filcfil 17556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-fin 6883  df-fi 7181  df-fbas 17536  df-fg 17537  df-fil 17557
  Copyright terms: Public domain W3C validator