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Theorem nrmr0reg 22450
 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.
StepHypRef Expression
1 nrmtop 22037 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
21adantr 485 . 2 ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Top)
3 simpll 767 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Nrm)
4 simprl 771 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥𝐽)
52adantr 485 . . . . . . 7 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Top)
6 toptopon2 21619 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
75, 6sylib 221 . . . . . 6 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ (TopOn‘ 𝐽))
8 simplr 769 . . . . . 6 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → (KQ‘𝐽) ∈ Fre)
9 simprr 773 . . . . . . 7 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦𝑥)
10 elunii 4804 . . . . . . 7 ((𝑦𝑥𝑥𝐽) → 𝑦 𝐽)
119, 4, 10syl2anc 588 . . . . . 6 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 𝐽)
12 eqid 2759 . . . . . . 7 (𝑧 𝐽 ↦ {𝑤𝐽𝑧𝑤}) = (𝑧 𝐽 ↦ {𝑤𝐽𝑧𝑤})
1312r0cld 22439 . . . . . 6 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝑦 𝐽) → {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ∈ (Clsd‘𝐽))
147, 8, 11, 13syl3anc 1369 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ∈ (Clsd‘𝐽))
15 simp1rr 1237 . . . . . . 7 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽 ∧ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)) → 𝑦𝑥)
164adantr 485 . . . . . . . . 9 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽) → 𝑥𝐽)
17 elequ2 2127 . . . . . . . . . . 11 (𝑏 = 𝑥 → (𝑎𝑏𝑎𝑥))
18 elequ2 2127 . . . . . . . . . . 11 (𝑏 = 𝑥 → (𝑦𝑏𝑦𝑥))
1917, 18bibi12d 350 . . . . . . . . . 10 (𝑏 = 𝑥 → ((𝑎𝑏𝑦𝑏) ↔ (𝑎𝑥𝑦𝑥)))
2019rspcv 3537 . . . . . . . . 9 (𝑥𝐽 → (∀𝑏𝐽 (𝑎𝑏𝑦𝑏) → (𝑎𝑥𝑦𝑥)))
2116, 20syl 17 . . . . . . . 8 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽) → (∀𝑏𝐽 (𝑎𝑏𝑦𝑏) → (𝑎𝑥𝑦𝑥)))
22213impia 1115 . . . . . . 7 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽 ∧ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)) → (𝑎𝑥𝑦𝑥))
2315, 22mpbird 260 . . . . . 6 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽 ∧ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)) → 𝑎𝑥)
2423rabssdv 3980 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑥)
25 nrmsep3 22056 . . . . 5 ((𝐽 ∈ Nrm ∧ (𝑥𝐽 ∧ {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ∈ (Clsd‘𝐽) ∧ {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑥)) → ∃𝑧𝐽 ({𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
263, 4, 14, 24, 25syl13anc 1370 . . . 4 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑧𝐽 ({𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
27 elequ1 2119 . . . . . . . . . 10 (𝑎 = 𝑦 → (𝑎𝑏𝑦𝑏))
2827bibi1d 348 . . . . . . . . 9 (𝑎 = 𝑦 → ((𝑎𝑏𝑦𝑏) ↔ (𝑦𝑏𝑦𝑏)))
2928ralbidv 3127 . . . . . . . 8 (𝑎 = 𝑦 → (∀𝑏𝐽 (𝑎𝑏𝑦𝑏) ↔ ∀𝑏𝐽 (𝑦𝑏𝑦𝑏)))
30 biidd 265 . . . . . . . . 9 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → (𝑦𝑏𝑦𝑏))
3130ralrimivw 3115 . . . . . . . 8 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → ∀𝑏𝐽 (𝑦𝑏𝑦𝑏))
3229, 11, 31elrabd 3605 . . . . . . 7 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 ∈ {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)})
33 ssel 3886 . . . . . . 7 ({𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑧 → (𝑦 ∈ {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} → 𝑦𝑧))
3432, 33syl5com 31 . . . . . 6 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → ({𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑧𝑦𝑧))
3534anim1d 614 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → (({𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥) → (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
3635reximdv 3198 . . . 4 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → (∃𝑧𝐽 ({𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥) → ∃𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
3726, 36mpd 15 . . 3 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
3837ralrimivva 3121 . 2 ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑥𝐽𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
39 isreg 22033 . 2 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
402, 38, 39sylanbrc 587 1 ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Reg)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  {crab 3075   ⊆ wss 3859  ∪ cuni 4799   ↦ cmpt 5113  ‘cfv 6336  Topctop 21594  TopOnctopon 21611  Clsdccld 21717  clsccl 21719  Frect1 22008  Regcreg 22010  Nrmcnrm 22011  KQckq 22394 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8419  df-qtop 16839  df-top 21595  df-topon 21612  df-cld 21720  df-cn 21928  df-t1 22015  df-reg 22017  df-nrm 22018  df-kq 22395 This theorem is referenced by:  nrmreg  22525
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