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Theorem regr1 23254
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 β†’ (KQβ€˜π½) ∈ Haus)

Proof of Theorem regr1
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 regtop 22837 . . 3 (𝐽 ∈ Reg β†’ 𝐽 ∈ Top)
2 toptopon2 22420 . . 3 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
31, 2sylib 217 . 2 (𝐽 ∈ Reg β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
4 eqid 2733 . . 3 (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦}) = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
54regr1lem2 23244 . 2 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ 𝐽 ∈ Reg) β†’ (KQβ€˜π½) ∈ Haus)
63, 5mpancom 687 1 (𝐽 ∈ Reg β†’ (KQβ€˜π½) ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107  {crab 3433  βˆͺ cuni 4909   ↦ cmpt 5232  β€˜cfv 6544  Topctop 22395  TopOnctopon 22412  Hauscha 22812  Regcreg 22813  KQckq 23197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-qtop 17453  df-top 22396  df-topon 22413  df-cld 22523  df-cls 22525  df-haus 22819  df-reg 22820  df-kq 23198
This theorem is referenced by:  reghaus  23329
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