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Theorem kqreg 20546
Description: The Kolmogorov quotient of a regular space is regular. By regr1 20545 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 20129 . . . 4  |-  ( J  e.  Reg  ->  J  e.  Top )
2 eqid 2404 . . . . 5  |-  U. J  =  U. J
32toptopon 19728 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
41, 3sylib 198 . . 3  |-  ( J  e.  Reg  ->  J  e.  (TopOn `  U. J ) )
5 eqid 2404 . . . 4  |-  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y }
)  =  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y }
)
65kqreglem1 20536 . . 3  |-  ( ( J  e.  (TopOn `  U. J )  /\  J  e.  Reg )  ->  (KQ `  J )  e.  Reg )
74, 6mpancom 669 . 2  |-  ( J  e.  Reg  ->  (KQ `  J )  e.  Reg )
8 regtop 20129 . . . . 5  |-  ( (KQ
`  J )  e. 
Reg  ->  (KQ `  J
)  e.  Top )
9 kqtop 20540 . . . . 5  |-  ( J  e.  Top  <->  (KQ `  J
)  e.  Top )
108, 9sylibr 214 . . . 4  |-  ( (KQ
`  J )  e. 
Reg  ->  J  e.  Top )
1110, 3sylib 198 . . 3  |-  ( (KQ
`  J )  e. 
Reg  ->  J  e.  (TopOn `  U. J ) )
125kqreglem2 20537 . . 3  |-  ( ( J  e.  (TopOn `  U. J )  /\  (KQ `  J )  e.  Reg )  ->  J  e.  Reg )
1311, 12mpancom 669 . 2  |-  ( (KQ
`  J )  e. 
Reg  ->  J  e.  Reg )
147, 13impbii 189 1  |-  ( J  e.  Reg  <->  (KQ `  J
)  e.  Reg )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    e. wcel 1844   {crab 2760   U.cuni 4193    |-> cmpt 4455   ` cfv 5571   Topctop 19688  TopOnctopon 19689   Regcreg 20105  KQckq 20488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-map 7461  df-qtop 15123  df-top 19693  df-topon 19696  df-cld 19814  df-cls 19816  df-cn 20023  df-reg 20112  df-kq 20489
This theorem is referenced by: (None)
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