Theorem nrmr0reg 22354

Description: A normal R₀ space is also regular. These spaces are usually referred to as normal regular spaces. (Contributed by Mario Carneiro, 25-Aug-2015.)

Assertion

Ref	Expression
nrmr0reg	⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Reg)

Proof of Theorem nrmr0reg

Dummy variables 𝑥 𝑦 𝑎 𝑏 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nrmtop 21941	. . 3 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2	1	adantr 484	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Top)
3		simpll 766	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Nrm)
4		simprl 770	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝐽)
5	2	adantr 484	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Top)
6		toptopon2 21523	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
7	5, 6	sylib 221	. . . . . 6 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
8		simplr 768	. . . . . 6 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (KQ‘𝐽) ∈ Fre)
9		simprr 772	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑥)
10		elunii 4805	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐽) → 𝑦 ∈ ∪ 𝐽)
11	9, 4, 10	syl2anc 587	. . . . . 6 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ ∪ 𝐽)
12		eqid 2798	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝐽 ↦ {𝑤 ∈ 𝐽 ∣ 𝑧 ∈ 𝑤}) = (𝑧 ∈ ∪ 𝐽 ↦ {𝑤 ∈ 𝐽 ∣ 𝑧 ∈ 𝑤})
13	12	r0cld 22343	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝑦 ∈ ∪ 𝐽) → {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ∈ (Clsd‘𝐽))
14	7, 8, 11, 13	syl3anc 1368	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ∈ (Clsd‘𝐽))
15		simp1rr 1236	. . . . . . 7 ⊢ ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ 𝑎 ∈ ∪ 𝐽 ∧ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)) → 𝑦 ∈ 𝑥)
16	4	adantr 484	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ 𝑎 ∈ ∪ 𝐽) → 𝑥 ∈ 𝐽)
17		elequ2 2126	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑥 → (𝑎 ∈ 𝑏 ↔ 𝑎 ∈ 𝑥))
18		elequ2 2126	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑥 → (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝑥))
19	17, 18	bibi12d 349	. . . . . . . . . 10 ⊢ (𝑏 = 𝑥 → ((𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏) ↔ (𝑎 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)))
20	19	rspcv 3566	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐽 → (∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏) → (𝑎 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)))
21	16, 20	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ 𝑎 ∈ ∪ 𝐽) → (∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏) → (𝑎 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)))
22	21	3impia 1114	. . . . . . 7 ⊢ ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ 𝑎 ∈ ∪ 𝐽 ∧ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)) → (𝑎 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
23	15, 22	mpbird 260	. . . . . 6 ⊢ ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ 𝑎 ∈ ∪ 𝐽 ∧ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)) → 𝑎 ∈ 𝑥)
24	23	rabssdv 4002	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ⊆ 𝑥)
25		nrmsep3 21960	. . . . 5 ⊢ ((𝐽 ∈ Nrm ∧ (𝑥 ∈ 𝐽 ∧ {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ∈ (Clsd‘𝐽) ∧ {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ⊆ 𝑥)) → ∃𝑧 ∈ 𝐽 ({𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
26	3, 4, 14, 24, 25	syl13anc 1369	. . . 4 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∃𝑧 ∈ 𝐽 ({𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
27		elequ1 2118	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏))
28	27	bibi1d 347	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → ((𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏) ↔ (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)))
29	28	ralbidv 3162	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → (∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏) ↔ ∀𝑏 ∈ 𝐽 (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)))
30		biidd 265	. . . . . . . . 9 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏))
31	30	ralrimivw 3150	. . . . . . . 8 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∀𝑏 ∈ 𝐽 (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏))
32	29, 11, 31	elrabd 3630	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)})
33		ssel 3908	. . . . . . 7 ⊢ ({𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ⊆ 𝑧 → (𝑦 ∈ {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} → 𝑦 ∈ 𝑧))
34	32, 33	syl5com 31	. . . . . 6 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ({𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ⊆ 𝑧 → 𝑦 ∈ 𝑧))
35	34	anim1d 613	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (({𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥) → (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
36	35	reximdv 3232	. . . 4 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (∃𝑧 ∈ 𝐽 ({𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥) → ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
37	26, 36	mpd 15	. . 3 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
38	37	ralrimivva 3156	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
39		isreg 21937	. 2 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
40	2, 38, 39	sylanbrc 586	1 ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Reg)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110   ⊆ wss 3881  ∪ cuni 4800   ↦ cmpt 5110  ‘cfv 6324  Topctop 21498  TopOnctopon 21515  Clsdccld 21621  clsccl 21623  Frect1 21912  Regcreg 21914  Nrmcnrm 21915  KQckq 22298

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-qtop 16772  df-top 21499  df-topon 21516  df-cld 21624  df-cn 21832  df-t1 21919  df-reg 21921  df-nrm 21922  df-kq 22299

This theorem is referenced by:  nrmreg  22429