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Theorem regr1 21534
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 21118 . . 3 (𝐽 ∈ Reg → 𝐽 ∈ Top)
2 eqid 2620 . . . 4 𝐽 = 𝐽
32toptopon 20703 . . 3 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
41, 3sylib 208 . 2 (𝐽 ∈ Reg → 𝐽 ∈ (TopOn‘ 𝐽))
5 eqid 2620 . . 3 (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})
65regr1lem2 21524 . 2 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus)
74, 6mpancom 702 1 (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1988  {crab 2913   cuni 4427  cmpt 4720  cfv 5876  Topctop 20679  TopOnctopon 20696  Hauscha 21093  Regcreg 21094  KQckq 21477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-qtop 16148  df-top 20680  df-topon 20697  df-cld 20804  df-cls 20806  df-haus 21100  df-reg 21101  df-kq 21478
This theorem is referenced by:  reghaus  21609
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