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Theorem nrmr0reg 21600
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 21188 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
21adantr 480 . 2 ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Top)
3 simpll 805 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Nrm)
4 simprl 809 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥𝐽)
52adantr 480 . . . . . . 7 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Top)
6 eqid 2651 . . . . . . . 8 𝐽 = 𝐽
76toptopon 20770 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
85, 7sylib 208 . . . . . 6 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ (TopOn‘ 𝐽))
9 simplr 807 . . . . . 6 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → (KQ‘𝐽) ∈ Fre)
10 simprr 811 . . . . . . 7 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦𝑥)
11 elunii 4473 . . . . . . 7 ((𝑦𝑥𝑥𝐽) → 𝑦 𝐽)
1210, 4, 11syl2anc 694 . . . . . 6 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 𝐽)
13 eqid 2651 . . . . . . 7 (𝑧 𝐽 ↦ {𝑤𝐽𝑧𝑤}) = (𝑧 𝐽 ↦ {𝑤𝐽𝑧𝑤})
1413r0cld 21589 . . . . . 6 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝑦 𝐽) → {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ∈ (Clsd‘𝐽))
158, 9, 12, 14syl3anc 1366 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ∈ (Clsd‘𝐽))
16 simp1rr 1147 . . . . . . 7 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽 ∧ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)) → 𝑦𝑥)
174adantr 480 . . . . . . . . 9 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽) → 𝑥𝐽)
18 elequ2 2044 . . . . . . . . . . 11 (𝑏 = 𝑥 → (𝑎𝑏𝑎𝑥))
19 elequ2 2044 . . . . . . . . . . 11 (𝑏 = 𝑥 → (𝑦𝑏𝑦𝑥))
2018, 19bibi12d 334 . . . . . . . . . 10 (𝑏 = 𝑥 → ((𝑎𝑏𝑦𝑏) ↔ (𝑎𝑥𝑦𝑥)))
2120rspcv 3336 . . . . . . . . 9 (𝑥𝐽 → (∀𝑏𝐽 (𝑎𝑏𝑦𝑏) → (𝑎𝑥𝑦𝑥)))
2217, 21syl 17 . . . . . . . 8 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽) → (∀𝑏𝐽 (𝑎𝑏𝑦𝑏) → (𝑎𝑥𝑦𝑥)))
23223impia 1280 . . . . . . 7 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽 ∧ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)) → (𝑎𝑥𝑦𝑥))
2416, 23mpbird 247 . . . . . 6 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽 ∧ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)) → 𝑎𝑥)
2524rabssdv 3715 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑥)
26 nrmsep3 21207 . . . . 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))
273, 4, 15, 25, 26syl13anc 1368 . . . 4 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑧𝐽 ({𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
28 biidd 252 . . . . . . . . 9 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → (𝑦𝑏𝑦𝑏))
2928ralrimivw 2996 . . . . . . . 8 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → ∀𝑏𝐽 (𝑦𝑏𝑦𝑏))
30 elequ1 2037 . . . . . . . . . . 11 (𝑎 = 𝑦 → (𝑎𝑏𝑦𝑏))
3130bibi1d 332 . . . . . . . . . 10 (𝑎 = 𝑦 → ((𝑎𝑏𝑦𝑏) ↔ (𝑦𝑏𝑦𝑏)))
3231ralbidv 3015 . . . . . . . . 9 (𝑎 = 𝑦 → (∀𝑏𝐽 (𝑎𝑏𝑦𝑏) ↔ ∀𝑏𝐽 (𝑦𝑏𝑦𝑏)))
3332elrab 3396 . . . . . . . 8 (𝑦 ∈ {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ↔ (𝑦 𝐽 ∧ ∀𝑏𝐽 (𝑦𝑏𝑦𝑏)))
3412, 29, 33sylanbrc 699 . . . . . . 7 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 ∈ {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)})
35 ssel 3630 . . . . . . 7 ({𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑧 → (𝑦 ∈ {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} → 𝑦𝑧))
3634, 35syl5com 31 . . . . . 6 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → ({𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑧𝑦𝑧))
3736anim1d 587 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → (({𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥) → (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
3837reximdv 3045 . . . 4 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → (∃𝑧𝐽 ({𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥) → ∃𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
3927, 38mpd 15 . . 3 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
4039ralrimivva 3000 . 2 ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑥𝐽𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
41 isreg 21184 . 2 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
422, 40, 41sylanbrc 699 1 ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Reg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054  wcel 2030  wral 2941  wrex 2942  {crab 2945  wss 3607   cuni 4468  cmpt 4762  cfv 5926  Topctop 20746  TopOnctopon 20763  Clsdccld 20868  clsccl 20870  Frect1 21159  Regcreg 21161  Nrmcnrm 21162  KQckq 21544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-qtop 16214  df-top 20747  df-topon 20764  df-cld 20871  df-cn 21079  df-t1 21166  df-reg 21168  df-nrm 21169  df-kq 21545
This theorem is referenced by:  nrmreg  21675
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