Theorem regr1 23728

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.

Step	Hyp	Ref	Expression
1		regtop 23311	. . 3 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top)
2		toptopon2 22896	. . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3	1, 2	sylib 218	. 2 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽))
4		eqid 2737	. . 3 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
5	4	regr1lem2 23718	. 2 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus)
6	3, 5	mpancom 689	1 ⊢ (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Haus)

Syntax hints:   → wi 4   ∈ wcel 2114  {crab 3390  ∪ cuni 4851   ↦ cmpt 5167  ‘cfv 6493  Topctop 22871  TopOnctopon 22888  Hauscha 23286  Regcreg 23287  KQckq 23671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-qtop 17465  df-top 22872  df-topon 22889  df-cld 22997  df-cls 22999  df-haus 23293  df-reg 23294  df-kq 23672

This theorem is referenced by:  reghaus  23803