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Theorem iscfilu 21227
 Description: The predicate " is a Cauchy filter base on uniform space ." (Contributed by Thierry Arnoux, 18-Nov-2017.)
Assertion
Ref Expression
iscfilu UnifOn CauFilu
Distinct variable groups:   ,,   ,
Allowed substitution hints:   ()   (,)

Proof of Theorem iscfilu
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elrnust 21163 . . . . 5 UnifOn UnifOn
2 unieq 4221 . . . . . . . . 9
32dmeqd 5048 . . . . . . . 8
43fveq2d 5876 . . . . . . 7
5 raleq 3023 . . . . . . 7
64, 5rabeqbidv 3073 . . . . . 6
7 df-cfilu 21226 . . . . . 6 CauFilu UnifOn
8 fvex 5882 . . . . . . 7
98rabex 4567 . . . . . 6
106, 7, 9fvmpt 5955 . . . . 5 UnifOn CauFilu
111, 10syl 17 . . . 4 UnifOn CauFilu
1211eleq2d 2490 . . 3 UnifOn CauFilu
13 rexeq 3024 . . . . 5
1413ralbidv 2862 . . . 4
1514elrab 3226 . . 3
1612, 15syl6bb 264 . 2 UnifOn CauFilu
17 ustbas2 21164 . . . . 5 UnifOn
1817fveq2d 5876 . . . 4 UnifOn
1918eleq2d 2490 . . 3 UnifOn
2019anbi1d 709 . 2 UnifOn
2116, 20bitr4d 259 1 UnifOn CauFilu
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437   wcel 1867  wral 2773  wrex 2774  crab 2777   wss 3433  cuni 4213   cxp 4843   cdm 4845   crn 4846  cfv 5592  cfbas 18886  UnifOncust 21138  CauFiluccfilu 21225 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-iota 5556  df-fun 5594  df-fn 5595  df-fv 5600  df-ust 21139  df-cfilu 21226 This theorem is referenced by:  cfilufbas  21228  cfiluexsm  21229  fmucnd  21231  cfilufg  21232  trcfilu  21233  cfiluweak  21234  neipcfilu  21235  cfilucfil  21498
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