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Theorem cfiluweak 20561
Description: A Cauchy filter base is also a Cauchy filter base on any coarser uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)
Assertion
Ref Expression
cfiluweak  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  F  e.  (CauFilu `  U ) )

Proof of Theorem cfiluweak
Dummy variables  u  a  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trust 20495 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
213adant3 1016 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
3 simp3 998 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )
4 iscfilu 20554 . . . . . 6  |-  ( ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A )  ->  ( F  e.  (CauFilu `  ( Ut  ( A  X.  A
) ) )  <->  ( F  e.  ( fBas `  A
)  /\  A. u  e.  ( Ut  ( A  X.  A ) ) E. a  e.  F  ( a  X.  a ) 
C_  u ) ) )
54biimpa 484 . . . . 5  |-  ( ( ( Ut  ( A  X.  A ) )  e.  (UnifOn `  A )  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  ( F  e.  ( fBas `  A
)  /\  A. u  e.  ( Ut  ( A  X.  A ) ) E. a  e.  F  ( a  X.  a ) 
C_  u ) )
62, 3, 5syl2anc 661 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  ( F  e.  ( fBas `  A )  /\  A. u  e.  ( Ut  ( A  X.  A ) ) E. a  e.  F  ( a  X.  a
)  C_  u )
)
76simpld 459 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  F  e.  ( fBas `  A
) )
8 fbsspw 20096 . . . . 5  |-  ( F  e.  ( fBas `  A
)  ->  F  C_  ~P A )
97, 8syl 16 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  F  C_ 
~P A )
10 simp2 997 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  A  C_  X )
11 sspwb 4696 . . . . 5  |-  ( A 
C_  X  <->  ~P A  C_ 
~P X )
1210, 11sylib 196 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  ~P A  C_  ~P X )
139, 12sstrd 3514 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  F  C_ 
~P X )
14 simp1 996 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  U  e.  (UnifOn `  X )
)
1514elfvexd 5894 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  X  e.  _V )
16 fbasweak 20129 . . 3  |-  ( ( F  e.  ( fBas `  A )  /\  F  C_ 
~P X  /\  X  e.  _V )  ->  F  e.  ( fBas `  X
) )
177, 13, 15, 16syl3anc 1228 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  F  e.  ( fBas `  X
) )
1814adantr 465 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  U  e.  (UnifOn `  X )
)
1915adantr 465 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  X  e.  _V )
2010adantr 465 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  A  C_  X )
2119, 20ssexd 4594 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  A  e.  _V )
22 xpexg 6586 . . . . . . 7  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
2321, 21, 22syl2anc 661 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  ( A  X.  A )  e. 
_V )
24 simpr 461 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  v  e.  U )
25 elrestr 14684 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  ( A  X.  A )  e. 
_V  /\  v  e.  U )  ->  (
v  i^i  ( A  X.  A ) )  e.  ( Ut  ( A  X.  A ) ) )
2618, 23, 24, 25syl3anc 1228 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  (
v  i^i  ( A  X.  A ) )  e.  ( Ut  ( A  X.  A ) ) )
276simprd 463 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  A. u  e.  ( Ut  ( A  X.  A ) ) E. a  e.  F  ( a  X.  a ) 
C_  u )
2827adantr 465 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  A. u  e.  ( Ut  ( A  X.  A ) ) E. a  e.  F  ( a  X.  a ) 
C_  u )
29 sseq2 3526 . . . . . . 7  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  (
( a  X.  a
)  C_  u  <->  ( a  X.  a )  C_  (
v  i^i  ( A  X.  A ) ) ) )
3029rexbidv 2973 . . . . . 6  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  ( E. a  e.  F  ( a  X.  a
)  C_  u  <->  E. a  e.  F  ( a  X.  a )  C_  (
v  i^i  ( A  X.  A ) ) ) )
3130rspcva 3212 . . . . 5  |-  ( ( ( v  i^i  ( A  X.  A ) )  e.  ( Ut  ( A  X.  A ) )  /\  A. u  e.  ( Ut  ( A  X.  A ) ) E. a  e.  F  ( a  X.  a ) 
C_  u )  ->  E. a  e.  F  ( a  X.  a
)  C_  ( v  i^i  ( A  X.  A
) ) )
3226, 28, 31syl2anc 661 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  E. a  e.  F  ( a  X.  a )  C_  (
v  i^i  ( A  X.  A ) ) )
33 inss1 3718 . . . . . 6  |-  ( v  i^i  ( A  X.  A ) )  C_  v
34 sstr 3512 . . . . . 6  |-  ( ( ( a  X.  a
)  C_  ( v  i^i  ( A  X.  A
) )  /\  (
v  i^i  ( A  X.  A ) )  C_  v )  ->  (
a  X.  a ) 
C_  v )
3533, 34mpan2 671 . . . . 5  |-  ( ( a  X.  a ) 
C_  ( v  i^i  ( A  X.  A
) )  ->  (
a  X.  a ) 
C_  v )
3635reximi 2932 . . . 4  |-  ( E. a  e.  F  ( a  X.  a ) 
C_  ( v  i^i  ( A  X.  A
) )  ->  E. a  e.  F  ( a  X.  a )  C_  v
)
3732, 36syl 16 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  E. a  e.  F  ( a  X.  a )  C_  v
)
3837ralrimiva 2878 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  A. v  e.  U  E. a  e.  F  ( a  X.  a )  C_  v
)
39 iscfilu 20554 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( F  e.  (CauFilu `  U )  <->  ( F  e.  ( fBas `  X
)  /\  A. v  e.  U  E. a  e.  F  ( a  X.  a )  C_  v
) ) )
40393ad2ant1 1017 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  ( F  e.  (CauFilu `  U
)  <->  ( F  e.  ( fBas `  X
)  /\  A. v  e.  U  E. a  e.  F  ( a  X.  a )  C_  v
) ) )
4117, 38, 40mpbir2and 920 1  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  F  e.  (CauFilu `  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ~Pcpw 4010    X. cxp 4997   ` cfv 5588  (class class class)co 6284   ↾t crest 14676   fBascfbas 18205  UnifOncust 20465  CauFiluccfilu 20552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-rest 14678  df-fbas 18215  df-ust 20466  df-cfilu 20553
This theorem is referenced by: (None)
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