Theorem nrmr0reg 23724

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 23311	. . 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 22893	. . . . . . 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 4856	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐽) → 𝑦 ∈ ∪ 𝐽)
11	9, 4, 10	syl2anc 585	. . . . . 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 23713	. . . . . 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 1374	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ∈ (Clsd‘𝐽))
15		simp1rr 1241	. . . . . . 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 2129	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑥 → (𝑎 ∈ 𝑏 ↔ 𝑎 ∈ 𝑥))
18		elequ2 2129	. . . . . . . . . . 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 3561	. . . . . . . . 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 4015	. . . . 5 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)} ⊆ 𝑥)
25		nrmsep3 23330	. . . . 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 1375	. . . 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 2121	. . . . . . . . . 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 3161	. . . . . . . 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 3134	. . . . . . . 8 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∀𝑏 ∈ 𝐽 (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏))
32	29, 11, 31	elrabd 3637	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ {𝑎 ∈ ∪ 𝐽 ∣ ∀𝑏 ∈ 𝐽 (𝑎 ∈ 𝑏 ↔ 𝑦 ∈ 𝑏)})
33		ssel 3916	. . . . . . 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 612	. . . . 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 3153	. . . 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 3181	. 2 ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
39		isreg 23307	. 2 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐽 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
40	2, 38, 39	sylanbrc 584	1 ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Reg)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3390   ⊆ wss 3890  ∪ cuni 4851   ↦ cmpt 5167  ‘cfv 6492  Topctop 22868  TopOnctopon 22885  Clsdccld 22991  clsccl 22993  Frect1 23282  Regcreg 23284  Nrmcnrm 23285  KQckq 23668

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8768  df-qtop 17462  df-top 22869  df-topon 22886  df-cld 22994  df-cn 23202  df-t1 23289  df-reg 23291  df-nrm 23292  df-kq 23669

This theorem is referenced by:  nrmreg  23799