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Theorem regr1 23774
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 23357 . . 3 (𝐽 ∈ Reg → 𝐽 ∈ Top)
2 toptopon2 22940 . . 3 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
31, 2sylib 218 . 2 (𝐽 ∈ Reg → 𝐽 ∈ (TopOn‘ 𝐽))
4 eqid 2735 . . 3 (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})
54regr1lem2 23764 . 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 2106  {crab 3433   cuni 4912  cmpt 5231  cfv 6563  Topctop 22915  TopOnctopon 22932  Hauscha 23332  Regcreg 23333  KQckq 23717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-qtop 17554  df-top 22916  df-topon 22933  df-cld 23043  df-cls 23045  df-haus 23339  df-reg 23340  df-kq 23718
This theorem is referenced by:  reghaus  23849
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