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Theorem cfiluweak 21092
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 21026 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
2 iscfilu 21085 . . . . . 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 ) ) )
32biimpa 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 ) )
41, 3stoic3 1632 . . . 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 )
)
54simpld 459 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  F  e.  ( fBas `  A
) )
6 fbsspw 20627 . . . . 5  |-  ( F  e.  ( fBas `  A
)  ->  F  C_  ~P A )
75, 6syl 17 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  F  C_ 
~P A )
8 simp2 1000 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  A  C_  X )
9 sspwb 4642 . . . . 5  |-  ( A 
C_  X  <->  ~P A  C_ 
~P X )
108, 9sylib 198 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  ~P A  C_  ~P X )
117, 10sstrd 3454 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  F  C_ 
~P X )
12 simp1 999 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  U  e.  (UnifOn `  X )
)
1312elfvexd 5879 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  X  e.  _V )
14 fbasweak 20660 . . 3  |-  ( ( F  e.  ( fBas `  A )  /\  F  C_ 
~P X  /\  X  e.  _V )  ->  F  e.  ( fBas `  X
) )
155, 11, 13, 14syl3anc 1232 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  ->  F  e.  ( fBas `  X
) )
1612adantr 465 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  U  e.  (UnifOn `  X )
)
1713adantr 465 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  X  e.  _V )
188adantr 465 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  A  C_  X )
1917, 18ssexd 4543 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  A  e.  _V )
20 xpexg 6586 . . . . . . 7  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
2119, 19, 20syl2anc 661 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  ( A  X.  A )  e. 
_V )
22 simpr 461 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  A  C_  X  /\  F  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )  /\  v  e.  U )  ->  v  e.  U )
23 elrestr 15045 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  ( A  X.  A )  e. 
_V  /\  v  e.  U )  ->  (
v  i^i  ( A  X.  A ) )  e.  ( Ut  ( A  X.  A ) ) )
2416, 21, 22, 23syl3anc 1232 . . . . 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 ) ) )
254simprd 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 )
2625adantr 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 )
27 sseq2 3466 . . . . . . 7  |-  ( u  =  ( v  i^i  ( A  X.  A
) )  ->  (
( a  X.  a
)  C_  u  <->  ( a  X.  a )  C_  (
v  i^i  ( A  X.  A ) ) ) )
2827rexbidv 2920 . . . . . 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 ) ) ) )
2928rspcva 3160 . . . . 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
) ) )
3024, 26, 29syl2anc 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 ) ) )
31 inss1 3661 . . . . . 6  |-  ( v  i^i  ( A  X.  A ) )  C_  v
32 sstr 3452 . . . . . 6  |-  ( ( ( a  X.  a
)  C_  ( v  i^i  ( A  X.  A
) )  /\  (
v  i^i  ( A  X.  A ) )  C_  v )  ->  (
a  X.  a ) 
C_  v )
3331, 32mpan2 671 . . . . 5  |-  ( ( a  X.  a ) 
C_  ( v  i^i  ( A  X.  A
) )  ->  (
a  X.  a ) 
C_  v )
3433reximi 2874 . . . 4  |-  ( E. a  e.  F  ( a  X.  a ) 
C_  ( v  i^i  ( A  X.  A
) )  ->  E. a  e.  F  ( a  X.  a )  C_  v
)
3530, 34syl 17 . . 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
)
3635ralrimiva 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
)
37 iscfilu 21085 . . 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
) ) )
38373ad2ant1 1020 . 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
) ) )
3915, 36, 38mpbir2and 925 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 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   A.wral 2756   E.wrex 2757   _Vcvv 3061    i^i cin 3415    C_ wss 3416   ~Pcpw 3957    X. cxp 4823   ` cfv 5571  (class class class)co 6280   ↾t crest 15037   fBascfbas 18728  UnifOncust 20996  CauFiluccfilu 21083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-rest 15039  df-fbas 18738  df-ust 20997  df-cfilu 21084
This theorem is referenced by: (None)
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