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Theorem kqreg 17458
Description: The Kolmogorov quotient of a regular space is regular. By regr1 17457 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 17077 . . . 4  |-  ( J  e.  Reg  ->  J  e.  Top )
2 eqid 2296 . . . . 5  |-  U. J  =  U. J
32toptopon 16687 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
41, 3sylib 188 . . 3  |-  ( J  e.  Reg  ->  J  e.  (TopOn `  U. J ) )
5 eqid 2296 . . . 4  |-  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y }
)  =  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y }
)
65kqreglem1 17448 . . 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 17077 . . . . 5  |-  ( (KQ
`  J )  e. 
Reg  ->  (KQ `  J
)  e.  Top )
9 kqtop 17452 . . . . 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 17449 . . 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 1696   {crab 2560   U.cuni 3843    e. cmpt 4093   ` cfv 5271   Topctop 16647  TopOnctopon 16648   Regcreg 17053  KQckq 17400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-qtop 13426  df-top 16652  df-topon 16655  df-cld 16772  df-cls 16774  df-cn 16973  df-reg 17060  df-kq 17401
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