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Theorem isreg 17396
 Description: The predicate "is a regular space." In a regular space, any open neighborhood has a closed subneighborhood. Note that some authors require the space to be Hausdorff (which would make it the same as T3), but we reserve the phrase "regular Hausdorff" for that as many topologists do. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
isreg
Distinct variable group:   ,,,

Proof of Theorem isreg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5728 . . . . . . . 8
21fveq1d 5730 . . . . . . 7
32sseq1d 3375 . . . . . 6
43anbi2d 685 . . . . 5
54rexeqbi1dv 2913 . . . 4
65ralbidv 2725 . . 3
76raleqbi1dv 2912 . 2
8 df-reg 17380 . 2
97, 8elrab2 3094 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725  wral 2705  wrex 2706   wss 3320  cfv 5454  ctop 16958  ccl 17082  creg 17373 This theorem is referenced by:  regtop  17397  regsep  17398  isreg2  17441  kqreglem1  17773  kqreglem2  17774  nrmr0reg  17781  reghmph  17825  utopreg  18282 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-reg 17380
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