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Theorem nrmr0reg 22275
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 21862 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
21adantr 481 . 2 ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Top)
3 simpll 763 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Nrm)
4 simprl 767 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥𝐽)
52adantr 481 . . . . . . 7 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Top)
6 toptopon2 21444 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
75, 6sylib 219 . . . . . 6 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ (TopOn‘ 𝐽))
8 simplr 765 . . . . . 6 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → (KQ‘𝐽) ∈ Fre)
9 simprr 769 . . . . . . 7 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦𝑥)
10 elunii 4841 . . . . . . 7 ((𝑦𝑥𝑥𝐽) → 𝑦 𝐽)
119, 4, 10syl2anc 584 . . . . . 6 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 𝐽)
12 eqid 2825 . . . . . . 7 (𝑧 𝐽 ↦ {𝑤𝐽𝑧𝑤}) = (𝑧 𝐽 ↦ {𝑤𝐽𝑧𝑤})
1312r0cld 22264 . . . . . 6 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝑦 𝐽) → {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ∈ (Clsd‘𝐽))
147, 8, 11, 13syl3anc 1365 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ∈ (Clsd‘𝐽))
15 simp1rr 1233 . . . . . . 7 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽 ∧ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)) → 𝑦𝑥)
164adantr 481 . . . . . . . . 9 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽) → 𝑥𝐽)
17 elequ2 2122 . . . . . . . . . . 11 (𝑏 = 𝑥 → (𝑎𝑏𝑎𝑥))
18 elequ2 2122 . . . . . . . . . . 11 (𝑏 = 𝑥 → (𝑦𝑏𝑦𝑥))
1917, 18bibi12d 347 . . . . . . . . . 10 (𝑏 = 𝑥 → ((𝑎𝑏𝑦𝑏) ↔ (𝑎𝑥𝑦𝑥)))
2019rspcv 3621 . . . . . . . . 9 (𝑥𝐽 → (∀𝑏𝐽 (𝑎𝑏𝑦𝑏) → (𝑎𝑥𝑦𝑥)))
2116, 20syl 17 . . . . . . . 8 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽) → (∀𝑏𝐽 (𝑎𝑏𝑦𝑏) → (𝑎𝑥𝑦𝑥)))
22213impia 1111 . . . . . . 7 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽 ∧ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)) → (𝑎𝑥𝑦𝑥))
2315, 22mpbird 258 . . . . . 6 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽 ∧ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)) → 𝑎𝑥)
2423rabssdv 4054 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑥)
25 nrmsep3 21881 . . . . 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 1366 . . . 4 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑧𝐽 ({𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
27 elequ1 2114 . . . . . . . . . 10 (𝑎 = 𝑦 → (𝑎𝑏𝑦𝑏))
2827bibi1d 345 . . . . . . . . 9 (𝑎 = 𝑦 → ((𝑎𝑏𝑦𝑏) ↔ (𝑦𝑏𝑦𝑏)))
2928ralbidv 3201 . . . . . . . 8 (𝑎 = 𝑦 → (∀𝑏𝐽 (𝑎𝑏𝑦𝑏) ↔ ∀𝑏𝐽 (𝑦𝑏𝑦𝑏)))
30 biidd 263 . . . . . . . . 9 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → (𝑦𝑏𝑦𝑏))
3130ralrimivw 3187 . . . . . . . 8 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → ∀𝑏𝐽 (𝑦𝑏𝑦𝑏))
3229, 11, 31elrabd 3685 . . . . . . 7 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 ∈ {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)})
33 ssel 3964 . . . . . . 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 610 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → (({𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥) → (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
3635reximdv 3277 . . . 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 3195 . 2 ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑥𝐽𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
39 isreg 21858 . 2 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
402, 38, 39sylanbrc 583 1 ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Reg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081  wcel 2107  wral 3142  wrex 3143  {crab 3146  wss 3939   cuni 4836  cmpt 5142  cfv 6351  Topctop 21419  TopOnctopon 21436  Clsdccld 21542  clsccl 21544  Frect1 21833  Regcreg 21835  Nrmcnrm 21836  KQckq 22219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8401  df-qtop 16772  df-top 21420  df-topon 21437  df-cld 21545  df-cn 21753  df-t1 21840  df-reg 21842  df-nrm 21843  df-kq 22220
This theorem is referenced by:  nrmreg  22350
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