Theorem nrmr0reg 23260

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 22847	. . 3 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2	1	adantr 481	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Top)
3		simpll 765	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Nrm)
4		simprl 769	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝐽)
5	2	adantr 481	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Top)
6		toptopon2 22427	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
7	5, 6	sylib 217	. . . . . 6 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
8		simplr 767	. . . . . 6 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (KQ‘𝐽) ∈ Fre)
9		simprr 771	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑥)
10		elunii 4913	. . . . . . 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 2732	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝐽 ↦ {𝑤 ∈ 𝐽 ∣ 𝑧 ∈ 𝑤}) = (𝑧 ∈ ∪ 𝐽 ↦ {𝑤 ∈ 𝐽 ∣ 𝑧 ∈ 𝑤})
13	12	r0cld 23249	. . . . . 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 1371	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ∈ (Clsd‘𝐽))
15		simp1rr 1239	. . . . . . 7 ⊢ ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ 𝑎 ∈ ∪ 𝐽 ∧ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)) → 𝑦 ∈ 𝑥)
16	4	adantr 481	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ 𝑎 ∈ ∪ 𝐽) → 𝑥 ∈ 𝐽)
17		elequ2 2121	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑥 → (𝑎 ∈ 𝑏 ↔ 𝑎 ∈ 𝑥))
18		elequ2 2121	. . . . . . . . . . 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 3608	. . . . . . . . 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 1117	. . . . . . 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 256	. . . . . 6 ⊢ ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ 𝑎 ∈ ∪ 𝐽 ∧ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)) → 𝑎 ∈ 𝑥)
24	23	rabssdv 4072	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ⊆ 𝑥)
25		nrmsep3 22866	. . . . 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 1372	. . . 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 2113	. . . . . . . . . 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 3177	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → (∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏) ↔ ∀𝑏 ∈ 𝐽 (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)))
30		biidd 261	. . . . . . . . 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 3685	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)})
33		ssel 3975	. . . . . . 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 3200	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
39		isreg 22843	. 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 205   ∧ wa 396   ∧ w3a 1087   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  {crab 3432   ⊆ wss 3948  ∪ cuni 4908   ↦ cmpt 5231  ‘cfv 6543  Topctop 22402  TopOnctopon 22419  Clsdccld 22527  clsccl 22529  Frect1 22818  Regcreg 22820  Nrmcnrm 22821  KQckq 23204

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-qtop 17455  df-top 22403  df-topon 22420  df-cld 22530  df-cn 22738  df-t1 22825  df-reg 22827  df-nrm 22828  df-kq 23205

This theorem is referenced by:  nrmreg  23335