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Theorem cfiluweak 19892
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 19826 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
213adant3 1008 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
3 simp3 990 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )
4 iscfilu 19885 . . . . . 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 19427 . . . . 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 989 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  A  C_  X )
11 sspwb 4562 . . . . 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 3387 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  F  C_ 
~P X )
14 simp1 988 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  U  e.  (UnifOn `  X )
)
1514elfvexd 5739 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  X  e.  _V )
16 fbasweak 19460 . . 3  |-  ( ( F  e.  ( fBas `  A )  /\  F  C_ 
~P X  /\  X  e.  _V )  ->  F  e.  ( fBas `  X
) )
177, 13, 15, 16syl3anc 1218 . 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 4460 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  A  e.  _V )
22 xpexg 6528 . . . . . . 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 14388 . . . . . 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 1218 . . . . 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 3399 . . . . . . 7  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  (
( a  X.  a
)  C_  u  <->  ( a  X.  a )  C_  (
v  i^i  ( A  X.  A ) ) ) )
3029rexbidv 2757 . . . . . 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 3092 . . . . 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 3591 . . . . . 6  |-  ( v  i^i  ( A  X.  A ) )  C_  v
34 sstr 3385 . . . . . 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 2844 . . . 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 2820 . 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 19885 . . 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 1009 . 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 913 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 965    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737   _Vcvv 2993    i^i cin 3348    C_ wss 3349   ~Pcpw 3881    X. cxp 4859   ` cfv 5439  (class class class)co 6112   ↾t crest 14380   fBascfbas 17826  UnifOncust 19796  CauFiluccfilu 19883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-rest 14382  df-fbas 17836  df-ust 19797  df-cfilu 19884
This theorem is referenced by: (None)
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