MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  regr1 Structured version   Unicode version

Theorem regr1 19983
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 19597 . . 3  |-  ( J  e.  Reg  ->  J  e.  Top )
2 eqid 2467 . . . 4  |-  U. J  =  U. J
32toptopon 19198 . . 3  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
41, 3sylib 196 . 2  |-  ( J  e.  Reg  ->  J  e.  (TopOn `  U. J ) )
5 eqid 2467 . . 3  |-  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y }
)  =  ( x  e.  U. J  |->  { y  e.  J  |  x  e.  y }
)
65regr1lem2 19973 . 2  |-  ( ( J  e.  (TopOn `  U. J )  /\  J  e.  Reg )  ->  (KQ `  J )  e.  Haus )
74, 6mpancom 669 1  |-  ( J  e.  Reg  ->  (KQ `  J )  e.  Haus )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   {crab 2818   U.cuni 4245    |-> cmpt 4505   ` cfv 5586   Topctop 19158  TopOnctopon 19159   Hauscha 19572   Regcreg 19573  KQckq 19926
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-qtop 14755  df-top 19163  df-topon 19166  df-cld 19283  df-cls 19285  df-haus 19579  df-reg 19580  df-kq 19927
This theorem is referenced by:  reghaus  20058
  Copyright terms: Public domain W3C validator