Theorem regr1 21925

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 21509	. . 3 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top)
2		eqid 2826	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	toptopon 21093	. . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
4	1, 3	sylib 210	. 2 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ (TopOn‘∪ 𝐽))
5		eqid 2826	. . 3 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
6	5	regr1lem2 21915	. 2 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Haus)
7	4, 6	mpancom 681	1 ⊢ (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Haus)

Syntax hints:   → wi 4   ∈ wcel 2166  {crab 3122  ∪ cuni 4659   ↦ cmpt 4953  ‘cfv 6124  Topctop 21069  TopOnctopon 21086  Hauscha 21484  Regcreg 21485  KQckq 21868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-iin 4744  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-qtop 16521  df-top 21070  df-topon 21087  df-cld 21195  df-cls 21197  df-haus 21491  df-reg 21492  df-kq 21869

This theorem is referenced by:  reghaus  22000