Theorem nrmr0reg 23757

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 23344	. . 3 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2	1	adantr 480	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Top)
3		simpll 767	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Nrm)
4		simprl 771	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝐽)
5	2	adantr 480	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Top)
6		toptopon2 22924	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
7	5, 6	sylib 218	. . . . . 6 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
8		simplr 769	. . . . . 6 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (KQ‘𝐽) ∈ Fre)
9		simprr 773	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑥)
10		elunii 4912	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐽) → 𝑦 ∈ ∪ 𝐽)
11	9, 4, 10	syl2anc 584	. . . . . 6 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ ∪ 𝐽)
12		eqid 2737	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝐽 ↦ {𝑤 ∈ 𝐽 ∣ 𝑧 ∈ 𝑤}) = (𝑧 ∈ ∪ 𝐽 ↦ {𝑤 ∈ 𝐽 ∣ 𝑧 ∈ 𝑤})
13	12	r0cld 23746	. . . . . 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 1373	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ∈ (Clsd‘𝐽))
15		simp1rr 1240	. . . . . . 7 ⊢ ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ 𝑎 ∈ ∪ 𝐽 ∧ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)) → 𝑦 ∈ 𝑥)
16	4	adantr 480	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ 𝑎 ∈ ∪ 𝐽) → 𝑥 ∈ 𝐽)
17		elequ2 2123	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑥 → (𝑎 ∈ 𝑏 ↔ 𝑎 ∈ 𝑥))
18		elequ2 2123	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑥 → (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝑥))
19	17, 18	bibi12d 345	. . . . . . . . . 10 ⊢ (𝑏 = 𝑥 → ((𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏) ↔ (𝑎 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)))
20	19	rspcv 3618	. . . . . . . . 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 1118	. . . . . . 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 257	. . . . . 6 ⊢ ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ 𝑎 ∈ ∪ 𝐽 ∧ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)) → 𝑎 ∈ 𝑥)
24	23	rabssdv 4075	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ⊆ 𝑥)
25		nrmsep3 23363	. . . . 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 1374	. . . 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 2115	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏))
28	27	bibi1d 343	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → ((𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏) ↔ (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)))
29	28	ralbidv 3178	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → (∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏) ↔ ∀𝑏 ∈ 𝐽 (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)))
30		biidd 262	. . . . . . . . 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 3694	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)})
33		ssel 3977	. . . . . . 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 611	. . . . 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 3170	. . . 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 3202	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
39		isreg 23340	. 2 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
40	2, 38, 39	sylanbrc 583	1 ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Reg)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  {crab 3436   ⊆ wss 3951  ∪ cuni 4907   ↦ cmpt 5225  ‘cfv 6561  Topctop 22899  TopOnctopon 22916  Clsdccld 23024  clsccl 23026  Frect1 23315  Regcreg 23317  Nrmcnrm 23318  KQckq 23701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-qtop 17552  df-top 22900  df-topon 22917  df-cld 23027  df-cn 23235  df-t1 23322  df-reg 23324  df-nrm 23325  df-kq 23702

This theorem is referenced by:  nrmreg  23832