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Theorem kqreg 19987
Description: The Kolmogorov quotient of a regular space is regular. By regr1 19986 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 19600 . . . 4  |-  ( J  e.  Reg  ->  J  e.  Top )
2 eqid 2467 . . . . 5  |-  U. J  =  U. J
32toptopon 19201 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
41, 3sylib 196 . . 3  |-  ( J  e.  Reg  ->  J  e.  (TopOn `  U. J ) )
5 eqid 2467 . . . 4  |-  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y }
)  =  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y }
)
65kqreglem1 19977 . . 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 19600 . . . . 5  |-  ( (KQ
`  J )  e. 
Reg  ->  (KQ `  J
)  e.  Top )
9 kqtop 19981 . . . . 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 19978 . . 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 1767   {crab 2818   U.cuni 4245    |-> cmpt 4505   ` cfv 5586   Topctop 19161  TopOnctopon 19162   Regcreg 19576  KQckq 19929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-qtop 14758  df-top 19166  df-topon 19169  df-cld 19286  df-cls 19288  df-cn 19494  df-reg 19583  df-kq 19930
This theorem is referenced by: (None)
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