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Theorem regr1 17784
Description: A regular space is R1, which means that any two topologically distinct points can be separated by neighborhoods. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
regr1  |-  ( J  e.  Reg  ->  (KQ `  J )  e.  Haus )

Proof of Theorem regr1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 regtop 17399 . . 3  |-  ( J  e.  Reg  ->  J  e.  Top )
2 eqid 2438 . . . 4  |-  U. J  =  U. J
32toptopon 17000 . . 3  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
41, 3sylib 190 . 2  |-  ( J  e.  Reg  ->  J  e.  (TopOn `  U. J ) )
5 eqid 2438 . . 3  |-  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y }
)  =  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y }
)
65regr1lem2 17774 . 2  |-  ( ( J  e.  (TopOn `  U. J )  /\  J  e.  Reg )  ->  (KQ `  J )  e.  Haus )
74, 6mpancom 652 1  |-  ( J  e.  Reg  ->  (KQ `  J )  e.  Haus )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726   {crab 2711   U.cuni 4017    e. cmpt 4268   ` cfv 5456   Topctop 16960  TopOnctopon 16961   Hauscha 17374   Regcreg 17375  KQckq 17727
This theorem is referenced by:  reghaus  17859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-qtop 13735  df-top 16965  df-topon 16968  df-cld 17085  df-cls 17087  df-haus 17381  df-reg 17382  df-kq 17728
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