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Theorem trcfilu 21358
Description: Condition for the trace of a Cauchy filter base to be a Cauchy filter base for the restricted uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)
Assertion
Ref Expression
trcfilu  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  -> 
( Ft  A )  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )

Proof of Theorem trcfilu
Dummy variables  a 
b  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1014 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  ->  U  e.  (UnifOn `  X
) )
2 simp2l 1040 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  ->  F  e.  (CauFilu `  U
) )
3 iscfilu 21352 . . . . . 6  |-  ( 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
) ) )
43biimpa 491 . . . . 5  |-  ( ( 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
) )
51, 2, 4syl2anc 671 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  -> 
( F  e.  (
fBas `  X )  /\  A. v  e.  U  E. a  e.  F  ( a  X.  a
)  C_  v )
)
65simpld 465 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  ->  F  e.  ( fBas `  X ) )
7 simp3 1016 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  ->  A  C_  X )
8 simp2r 1041 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  ->  -.  (/)  e.  ( Ft  A ) )
9 trfbas2 20907 . . . 4  |-  ( ( F  e.  ( fBas `  X )  /\  A  C_  X )  ->  (
( Ft  A )  e.  (
fBas `  A )  <->  -.  (/)  e.  ( Ft  A ) ) )
109biimpar 492 . . 3  |-  ( ( ( F  e.  (
fBas `  X )  /\  A  C_  X )  /\  -.  (/)  e.  ( Ft  A ) )  -> 
( Ft  A )  e.  (
fBas `  A )
)
116, 7, 8, 10syl21anc 1275 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  -> 
( Ft  A )  e.  (
fBas `  A )
)
122ad5antr 745 . . . . . . 7  |-  ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  /\  v  e.  U )  /\  w  =  ( v  i^i  ( A  X.  A
) ) )  /\  a  e.  F )  /\  ( a  X.  a
)  C_  v )  ->  F  e.  (CauFilu `  U
) )
131adantr 471 . . . . . . . . . 10  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  ->  U  e.  (UnifOn `  X ) )
1413elfvexd 5916 . . . . . . . . 9  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  ->  X  e.  _V )
157adantr 471 . . . . . . . . 9  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  ->  A  C_  X
)
1614, 15ssexd 4564 . . . . . . . 8  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  ->  A  e.  _V )
1716ad4antr 743 . . . . . . 7  |-  ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  /\  v  e.  U )  /\  w  =  ( v  i^i  ( A  X.  A
) ) )  /\  a  e.  F )  /\  ( a  X.  a
)  C_  v )  ->  A  e.  _V )
18 simplr 767 . . . . . . 7  |-  ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  /\  v  e.  U )  /\  w  =  ( v  i^i  ( A  X.  A
) ) )  /\  a  e.  F )  /\  ( a  X.  a
)  C_  v )  ->  a  e.  F )
19 elrestr 15376 . . . . . . 7  |-  ( ( F  e.  (CauFilu `  U
)  /\  A  e.  _V  /\  a  e.  F
)  ->  ( a  i^i  A )  e.  ( Ft  A ) )
2012, 17, 18, 19syl3anc 1276 . . . . . 6  |-  ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  /\  v  e.  U )  /\  w  =  ( v  i^i  ( A  X.  A
) ) )  /\  a  e.  F )  /\  ( a  X.  a
)  C_  v )  ->  ( a  i^i  A
)  e.  ( Ft  A ) )
21 inxp 4986 . . . . . . 7  |-  ( ( a  X.  a )  i^i  ( A  X.  A ) )  =  ( ( a  i^i 
A )  X.  (
a  i^i  A )
)
22 simpr 467 . . . . . . . . 9  |-  ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  /\  v  e.  U )  /\  w  =  ( v  i^i  ( A  X.  A
) ) )  /\  a  e.  F )  /\  ( a  X.  a
)  C_  v )  ->  ( a  X.  a
)  C_  v )
23 ssrin 3669 . . . . . . . . 9  |-  ( ( a  X.  a ) 
C_  v  ->  (
( a  X.  a
)  i^i  ( A  X.  A ) )  C_  ( v  i^i  ( A  X.  A ) ) )
2422, 23syl 17 . . . . . . . 8  |-  ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  /\  v  e.  U )  /\  w  =  ( v  i^i  ( A  X.  A
) ) )  /\  a  e.  F )  /\  ( a  X.  a
)  C_  v )  ->  ( ( a  X.  a )  i^i  ( A  X.  A ) ) 
C_  ( v  i^i  ( A  X.  A
) ) )
25 simpllr 774 . . . . . . . 8  |-  ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  /\  v  e.  U )  /\  w  =  ( v  i^i  ( A  X.  A
) ) )  /\  a  e.  F )  /\  ( a  X.  a
)  C_  v )  ->  w  =  ( v  i^i  ( A  X.  A ) ) )
2624, 25sseqtr4d 3481 . . . . . . 7  |-  ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  /\  v  e.  U )  /\  w  =  ( v  i^i  ( A  X.  A
) ) )  /\  a  e.  F )  /\  ( a  X.  a
)  C_  v )  ->  ( ( a  X.  a )  i^i  ( A  X.  A ) ) 
C_  w )
2721, 26syl5eqssr 3489 . . . . . 6  |-  ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  /\  v  e.  U )  /\  w  =  ( v  i^i  ( A  X.  A
) ) )  /\  a  e.  F )  /\  ( a  X.  a
)  C_  v )  ->  ( ( a  i^i 
A )  X.  (
a  i^i  A )
)  C_  w )
28 id 22 . . . . . . . . 9  |-  ( b  =  ( a  i^i 
A )  ->  b  =  ( a  i^i 
A ) )
2928sqxpeqd 4879 . . . . . . . 8  |-  ( b  =  ( a  i^i 
A )  ->  (
b  X.  b )  =  ( ( a  i^i  A )  X.  ( a  i^i  A
) ) )
3029sseq1d 3471 . . . . . . 7  |-  ( b  =  ( a  i^i 
A )  ->  (
( b  X.  b
)  C_  w  <->  ( (
a  i^i  A )  X.  ( a  i^i  A
) )  C_  w
) )
3130rspcev 3162 . . . . . 6  |-  ( ( ( a  i^i  A
)  e.  ( Ft  A )  /\  ( ( a  i^i  A )  X.  ( a  i^i 
A ) )  C_  w )  ->  E. b  e.  ( Ft  A ) ( b  X.  b )  C_  w )
3220, 27, 31syl2anc 671 . . . . 5  |-  ( ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  /\  v  e.  U )  /\  w  =  ( v  i^i  ( A  X.  A
) ) )  /\  a  e.  F )  /\  ( a  X.  a
)  C_  v )  ->  E. b  e.  ( Ft  A ) ( b  X.  b )  C_  w )
335simprd 469 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  ->  A. v  e.  U  E. a  e.  F  ( a  X.  a
)  C_  v )
3433r19.21bi 2769 . . . . . . 7  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  v  e.  U )  ->  E. a  e.  F  ( a  X.  a
)  C_  v )
35343ad2antr2 1180 . . . . . 6  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  ( w  e.  ( Ut  ( A  X.  A
) )  /\  v  e.  U  /\  w  =  ( v  i^i  ( A  X.  A
) ) ) )  ->  E. a  e.  F  ( a  X.  a
)  C_  v )
36353anassrs 1240 . . . . 5  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  ( F  e.  (CauFilu `  U )  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  /\  v  e.  U
)  /\  w  =  ( v  i^i  ( A  X.  A ) ) )  ->  E. a  e.  F  ( a  X.  a )  C_  v
)
3732, 36r19.29a 2944 . . . 4  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  ( F  e.  (CauFilu `  U )  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  /\  v  e.  U
)  /\  w  =  ( v  i^i  ( A  X.  A ) ) )  ->  E. b  e.  ( Ft  A ) ( b  X.  b )  C_  w )
38 xpexg 6620 . . . . . 6  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
3916, 16, 38syl2anc 671 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  ->  ( A  X.  A )  e.  _V )
40 simpr 467 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  ->  w  e.  ( Ut  ( A  X.  A ) ) )
41 elrest 15375 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  ( A  X.  A )  e. 
_V )  ->  (
w  e.  ( Ut  ( A  X.  A ) )  <->  E. v  e.  U  w  =  ( v  i^i  ( A  X.  A
) ) ) )
4241biimpa 491 . . . . 5  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( A  X.  A )  e. 
_V )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  ->  E. v  e.  U  w  =  ( v  i^i  ( A  X.  A
) ) )
4313, 39, 40, 42syl21anc 1275 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  ->  E. v  e.  U  w  =  ( v  i^i  ( A  X.  A ) ) )
4437, 43r19.29a 2944 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  /\  w  e.  ( Ut  ( A  X.  A ) ) )  ->  E. b  e.  ( Ft  A ) ( b  X.  b )  C_  w )
4544ralrimiva 2814 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  ->  A. w  e.  ( Ut  ( A  X.  A
) ) E. b  e.  ( Ft  A ) ( b  X.  b )  C_  w )
46 trust 21293 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
471, 7, 46syl2anc 671 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  -> 
( Ut  ( A  X.  A ) )  e.  (UnifOn `  A )
)
48 iscfilu 21352 . . 3  |-  ( ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A )  ->  (
( Ft  A )  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) )  <-> 
( ( Ft  A )  e.  ( fBas `  A
)  /\  A. w  e.  ( Ut  ( A  X.  A ) ) E. b  e.  ( Ft  A ) ( b  X.  b )  C_  w
) ) )
4947, 48syl 17 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  -> 
( ( Ft  A )  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) )  <->  ( ( Ft  A )  e.  ( fBas `  A )  /\  A. w  e.  ( Ut  ( A  X.  A ) ) E. b  e.  ( Ft  A ) ( b  X.  b )  C_  w ) ) )
5011, 45, 49mpbir2and 938 1  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  -> 
( Ft  A )  e.  (CauFilu `  ( Ut  ( A  X.  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   A.wral 2749   E.wrex 2750   _Vcvv 3057    i^i cin 3415    C_ wss 3416   (/)c0 3743    X. cxp 4851   ` cfv 5601  (class class class)co 6315   ↾t crest 15368   fBascfbas 19007  UnifOncust 21263  CauFiluccfilu 21350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-rest 15370  df-fbas 19016  df-ust 21264  df-cfilu 21351
This theorem is referenced by:  ucnextcn  21368
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