Theorem nrmr0reg 24061

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 23647	. . 3 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2	1	adantr 486	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Top)
3		simpll 779	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Nrm)
4		simprl 783	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝐽)
5	2	adantr 486	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Top)
6		toptopon2 23229	. . . . . . 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 781	. . . . . 6 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (KQ‘𝐽) ∈ Fre)
9		simprr 785	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑥)
10		elunii 4872	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐽) → 𝑦 ∈ ∪ 𝐽)
11	9, 4, 10	syl2anc 596	. . . . . 6 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ ∪ 𝐽)
12		eqid 2761	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝐽 ↦ {𝑤 ∈ 𝐽 ∣ 𝑧 ∈ 𝑤}) = (𝑧 ∈ ∪ 𝐽 ↦ {𝑤 ∈ 𝐽 ∣ 𝑧 ∈ 𝑤})
13	12	r0cld 24050	. . . . . 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 1398	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ∈ (Clsd‘𝐽))
15		simp1rr 1258	. . . . . . 7 ⊢ ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ 𝑎 ∈ ∪ 𝐽 ∧ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)) → 𝑦 ∈ 𝑥)
16	4	adantr 486	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ 𝑎 ∈ ∪ 𝐽) → 𝑥 ∈ 𝐽)
17		elequ2 2160	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑥 → (𝑎 ∈ 𝑏 ↔ 𝑎 ∈ 𝑥))
18		elequ2 2160	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑥 → (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝑥))
19	17, 18	bibi12d 348	. . . . . . . . . 10 ⊢ (𝑏 = 𝑥 → ((𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏) ↔ (𝑎 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)))
20	19	rspcv 3573	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐽 → (∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏) → (𝑎 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)))
21	16, 20	syl 18	. . . . . . . 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 1135	. . . . . . 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 4022	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ⊆ 𝑥)
25		nrmsep3 23666	. . . . 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 1399	. . . 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 2152	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏))
28	27	bibi1d 346	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → ((𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏) ↔ (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)))
29	28	ralbidv 3186	. . . . . . . 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 3159	. . . . . . . 8 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∀𝑏 ∈ 𝐽 (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏))
32	29, 11, 31	elrabd 3647	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)})
33		ssel 3925	. . . . . . 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 32	. . . . . 6 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ({𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ⊆ 𝑧 → 𝑦 ∈ 𝑧))
35	34	anim1d 623	. . . . 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 3178	. . . 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 16	. . 3 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
38	37	ralrimivva 3206	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
39		isreg 23643	. 2 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
40	2, 38, 39	sylanbrc 595	1 ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Reg)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {crab 3413   ⊆ wss 3899  ∪ cuni 4867   ↦ cmpt 5186  ‘cfv 6537  Topctop 23204  TopOnctopon 23221  Clsdccld 23327  clsccl 23329  Frect1 23618  Regcreg 23620  Nrmcnrm 23621  KQckq 24005

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-map 8842  df-qtop 17672  df-top 23205  df-topon 23222  df-cld 23330  df-cn 23538  df-t1 23625  df-reg 23627  df-nrm 23628  df-kq 24006

This theorem is used by:  nrmreg  24136