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Theorem regr1 23759
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 ∈ Reg → (KQ‘𝐽) ∈ Haus)

Proof of Theorem regr1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 regtop 23342 . . 3 (𝐽 ∈ Reg → 𝐽 ∈ Top)
2 toptopon2 22925 . . 3 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
31, 2sylib 218 . 2 (𝐽 ∈ Reg → 𝐽 ∈ (TopOn‘ 𝐽))
4 eqid 2736 . . 3 (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})
54regr1lem2 23749 . 2 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus)
63, 5mpancom 688 1 (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  {crab 3435   cuni 4906  cmpt 5224  cfv 6560  Topctop 22900  TopOnctopon 22917  Hauscha 23317  Regcreg 23318  KQckq 23702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-qtop 17553  df-top 22901  df-topon 22918  df-cld 23028  df-cls 23030  df-haus 23324  df-reg 23325  df-kq 23703
This theorem is referenced by:  reghaus  23834
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