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Theorem cfilufg 20531
 Description: The filter generated by a Cauchy filter base is still a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.)
Assertion
Ref Expression
cfilufg UnifOn CauFilu CauFilu

Proof of Theorem cfilufg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfilufbas 20527 . . 3 UnifOn CauFilu
2 fgcl 20114 . . 3
3 filfbas 20084 . . 3
41, 2, 33syl 20 . 2 UnifOn CauFilu
51ad3antrrr 729 . . . . . . 7 UnifOn CauFilu
6 ssfg 20108 . . . . . . 7
75, 6syl 16 . . . . . 6 UnifOn CauFilu
8 simplr 754 . . . . . 6 UnifOn CauFilu
97, 8sseldd 3505 . . . . 5 UnifOn CauFilu
10 id 22 . . . . . . . 8
1110, 10xpeq12d 5024 . . . . . . 7
1211sseq1d 3531 . . . . . 6
1312rspcev 3214 . . . . 5
149, 13sylancom 667 . . . 4 UnifOn CauFilu
15 iscfilu 20526 . . . . . 6 UnifOn CauFilu
1615simplbda 624 . . . . 5 UnifOn CauFilu
1716r19.21bi 2833 . . . 4 UnifOn CauFilu
1814, 17r19.29a 3003 . . 3 UnifOn CauFilu
1918ralrimiva 2878 . 2 UnifOn CauFilu
20 iscfilu 20526 . . 3 UnifOn CauFilu
2120adantr 465 . 2 UnifOn CauFilu CauFilu
224, 19, 21mpbir2and 920 1 UnifOn CauFilu CauFilu
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wcel 1767  wral 2814  wrex 2815   wss 3476   cxp 4997  cfv 5586  (class class class)co 6282  cfbas 18177  cfg 18178  cfil 20081  UnifOncust 20437  CauFiluccfilu 20524 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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574 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-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-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 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-fbas 18187  df-fg 18188  df-fil 20082  df-ust 20438  df-cfilu 20525 This theorem is referenced by:  ucnextcn  20542
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