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Theorem kqreg 17442
Description: The Kolmogorov quotient of a regular space is regular. By regr1 17441 it is also Hausdorff, so we can also say that a space is regular iff the Kolmogorov quotient is regular Hausdorff (T3). (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
kqreg  |-  ( J  e.  Reg  <->  (KQ `  J
)  e.  Reg )

Proof of Theorem kqreg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 regtop 17061 . . . 4  |-  ( J  e.  Reg  ->  J  e.  Top )
2 eqid 2283 . . . . 5  |-  U. J  =  U. J
32toptopon 16671 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
41, 3sylib 188 . . 3  |-  ( J  e.  Reg  ->  J  e.  (TopOn `  U. J ) )
5 eqid 2283 . . . 4  |-  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y }
)  =  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y }
)
65kqreglem1 17432 . . 3  |-  ( ( J  e.  (TopOn `  U. J )  /\  J  e.  Reg )  ->  (KQ `  J )  e.  Reg )
74, 6mpancom 650 . 2  |-  ( J  e.  Reg  ->  (KQ `  J )  e.  Reg )
8 regtop 17061 . . . . 5  |-  ( (KQ
`  J )  e. 
Reg  ->  (KQ `  J
)  e.  Top )
9 kqtop 17436 . . . . 5  |-  ( J  e.  Top  <->  (KQ `  J
)  e.  Top )
108, 9sylibr 203 . . . 4  |-  ( (KQ
`  J )  e. 
Reg  ->  J  e.  Top )
1110, 3sylib 188 . . 3  |-  ( (KQ
`  J )  e. 
Reg  ->  J  e.  (TopOn `  U. J ) )
125kqreglem2 17433 . . 3  |-  ( ( J  e.  (TopOn `  U. J )  /\  (KQ `  J )  e.  Reg )  ->  J  e.  Reg )
1311, 12mpancom 650 . 2  |-  ( (KQ
`  J )  e. 
Reg  ->  J  e.  Reg )
147, 13impbii 180 1  |-  ( J  e.  Reg  <->  (KQ `  J
)  e.  Reg )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1684   {crab 2547   U.cuni 3827    e. cmpt 4077   ` cfv 5255   Topctop 16631  TopOnctopon 16632   Regcreg 17037  KQckq 17384
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-qtop 13410  df-top 16636  df-topon 16639  df-cld 16756  df-cls 16758  df-cn 16957  df-reg 17044  df-kq 17385
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