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Theorem trcfilu 20624
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 996 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  ->  U  e.  (UnifOn `  X
) )
2 simp2l 1022 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  ->  F  e.  (CauFilu `  U
) )
3 iscfilu 20618 . . . . . 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 484 . . . . 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 661 . . . 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 459 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  ->  F  e.  ( fBas `  X ) )
7 simp3 998 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  ->  A  C_  X )
8 simp2r 1023 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  ->  -.  (/)  e.  ( Ft  A ) )
9 trfbas2 20171 . . . 4  |-  ( ( F  e.  ( fBas `  X )  /\  A  C_  X )  ->  (
( Ft  A )  e.  (
fBas `  A )  <->  -.  (/)  e.  ( Ft  A ) ) )
109biimpar 485 . . 3  |-  ( ( ( F  e.  (
fBas `  X )  /\  A  C_  X )  /\  -.  (/)  e.  ( Ft  A ) )  -> 
( Ft  A )  e.  (
fBas `  A )
)
116, 7, 8, 10syl21anc 1227 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  -> 
( Ft  A )  e.  (
fBas `  A )
)
122ad5antr 733 . . . . . . 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 465 . . . . . . . . . 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 5894 . . . . . . . . 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 465 . . . . . . . . 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 4594 . . . . . . . 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 731 . . . . . . 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 754 . . . . . . 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 14687 . . . . . . 7  |-  ( ( F  e.  (CauFilu `  U
)  /\  A  e.  _V  /\  a  e.  F
)  ->  ( a  i^i  A )  e.  ( Ft  A ) )
2012, 17, 18, 19syl3anc 1228 . . . . . 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 5135 . . . . . . 7  |-  ( ( a  X.  a )  i^i  ( A  X.  A ) )  =  ( ( a  i^i 
A )  X.  (
a  i^i  A )
)
22 simpr 461 . . . . . . . . 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 3723 . . . . . . . . 9  |-  ( ( a  X.  a ) 
C_  v  ->  (
( a  X.  a
)  i^i  ( A  X.  A ) )  C_  ( v  i^i  ( A  X.  A ) ) )
2422, 23syl 16 . . . . . . . 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 758 . . . . . . . 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 3541 . . . . . . 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 3549 . . . . . 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 ) )
2928, 28xpeq12d 5024 . . . . . . . 8  |-  ( b  =  ( a  i^i 
A )  ->  (
b  X.  b )  =  ( ( a  i^i  A )  X.  ( a  i^i  A
) ) )
3029sseq1d 3531 . . . . . . 7  |-  ( b  =  ( a  i^i 
A )  ->  (
( b  X.  b
)  C_  w  <->  ( (
a  i^i  A )  X.  ( a  i^i  A
) )  C_  w
) )
3130rspcev 3214 . . . . . 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 661 . . . . 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 463 . . . . . . . 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 2833 . . . . . . 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 1162 . . . . . 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 1218 . . . . 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 3003 . . . 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 6587 . . . . . 6  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( A  X.  A
)  e.  _V )
3916, 16, 38syl2anc 661 . . . . 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 461 . . . . 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 14686 . . . . . 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 484 . . . . 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 1227 . . . 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 3003 . . 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 2878 . 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 20559 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  A  C_  X )  ->  ( Ut  ( A  X.  A
) )  e.  (UnifOn `  A ) )
471, 7, 46syl2anc 661 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  ( F  e.  (CauFilu `  U
)  /\  -.  (/)  e.  ( Ft  A ) )  /\  A  C_  X )  -> 
( Ut  ( A  X.  A ) )  e.  (UnifOn `  A )
)
48 iscfilu 20618 . . 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 16 . 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 920 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 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   (/)c0 3785    X. cxp 4997   ` cfv 5588  (class class class)co 6285   ↾t crest 14679   fBascfbas 18217  UnifOncust 20529  CauFiluccfilu 20616
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 6577
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 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-rest 14681  df-fbas 18227  df-ust 20530  df-cfilu 20617
This theorem is referenced by:  ucnextcn  20634
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