MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kqreg Structured version   Unicode version

Theorem kqreg 19299
Description: The Kolmogorov quotient of a regular space is regular. By regr1 19298 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 18912 . . . 4  |-  ( J  e.  Reg  ->  J  e.  Top )
2 eqid 2438 . . . . 5  |-  U. J  =  U. J
32toptopon 18513 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
41, 3sylib 196 . . 3  |-  ( J  e.  Reg  ->  J  e.  (TopOn `  U. J ) )
5 eqid 2438 . . . 4  |-  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y }
)  =  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y }
)
65kqreglem1 19289 . . 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 18912 . . . . 5  |-  ( (KQ
`  J )  e. 
Reg  ->  (KQ `  J
)  e.  Top )
9 kqtop 19293 . . . . 5  |-  ( J  e.  Top  <->  (KQ `  J
)  e.  Top )
108, 9sylibr 212 . . . 4  |-  ( (KQ
`  J )  e. 
Reg  ->  J  e.  Top )
1110, 3sylib 196 . . 3  |-  ( (KQ
`  J )  e. 
Reg  ->  J  e.  (TopOn `  U. J ) )
125kqreglem2 19290 . . 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 188 1  |-  ( J  e.  Reg  <->  (KQ `  J
)  e.  Reg )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1756   {crab 2714   U.cuni 4086    e. cmpt 4345   ` cfv 5413   Topctop 18473  TopOnctopon 18474   Regcreg 18888  KQckq 19241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-map 7208  df-qtop 14437  df-top 18478  df-topon 18481  df-cld 18598  df-cls 18600  df-cn 18806  df-reg 18895  df-kq 19242
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator