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Theorem cnfilca 25556
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 17550 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  X  e.  F )
2 ssexg 4160 . . . . . . 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 3754 . . . . . 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 3349 . . . . 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 2564 . 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 17561 . . . . . . 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 17527 . . . . . 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 17546 . . . . 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 3351 . . 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 3487 . . . 4  |-  ( ( A  C_  X  /\  A  =/=  (/) )  ->  X  =/=  (/) )
25243adant1 973 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X  /\  A  =/=  (/) )  ->  X  =/=  (/) )
26 efilcp 25552 . . 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 17543 . . . . 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 17564 . . . 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 3363 . . . . . . . 8  |-  ( y  =  A  ->  (
y  i^i  x )  =  ( A  i^i  x ) )
34 incom 3361 . . . . . . . 8  |-  ( A  i^i  x )  =  ( x  i^i  A
)
3533, 34syl6eq 2331 . . . . . . 7  |-  ( y  =  A  ->  (
y  i^i  x )  =  ( x  i^i 
A ) )
3635neeq1d 2459 . . . . . 6  |-  ( y  =  A  ->  (
( y  i^i  x
)  =/=  (/)  <->  ( x  i^i  A )  =/=  (/) ) )
3736ralbidv 2563 . . . . 5  |-  ( y  =  A  ->  ( A. x  e.  F  ( y  i^i  x
)  =/=  (/)  <->  A. x  e.  F  ( x  i^i  A )  =/=  (/) ) )
3837ralsng 3672 . . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   ` cfv 5255   ficfi 7164   fBascfbas 17518   Filcfil 17540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-fi 7165  df-fbas 17520  df-fg 17521  df-fil 17541
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