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Theorem nrmr0reg 21465
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 21053 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
21adantr 481 . 2 ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Top)
3 simpll 789 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Nrm)
4 simprl 793 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥𝐽)
52adantr 481 . . . . . . 7 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Top)
6 eqid 2621 . . . . . . . 8 𝐽 = 𝐽
76toptopon 20647 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
85, 7sylib 208 . . . . . 6 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ (TopOn‘ 𝐽))
9 simplr 791 . . . . . 6 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → (KQ‘𝐽) ∈ Fre)
10 simprr 795 . . . . . . 7 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦𝑥)
11 elunii 4409 . . . . . . 7 ((𝑦𝑥𝑥𝐽) → 𝑦 𝐽)
1210, 4, 11syl2anc 692 . . . . . 6 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 𝐽)
13 eqid 2621 . . . . . . 7 (𝑧 𝐽 ↦ {𝑤𝐽𝑧𝑤}) = (𝑧 𝐽 ↦ {𝑤𝐽𝑧𝑤})
1413r0cld 21454 . . . . . 6 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝑦 𝐽) → {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ∈ (Clsd‘𝐽))
158, 9, 12, 14syl3anc 1323 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ∈ (Clsd‘𝐽))
16 simp1rr 1125 . . . . . . 7 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽 ∧ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)) → 𝑦𝑥)
174adantr 481 . . . . . . . . 9 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽) → 𝑥𝐽)
18 elequ2 2001 . . . . . . . . . . 11 (𝑏 = 𝑥 → (𝑎𝑏𝑎𝑥))
19 elequ2 2001 . . . . . . . . . . 11 (𝑏 = 𝑥 → (𝑦𝑏𝑦𝑥))
2018, 19bibi12d 335 . . . . . . . . . 10 (𝑏 = 𝑥 → ((𝑎𝑏𝑦𝑏) ↔ (𝑎𝑥𝑦𝑥)))
2120rspcv 3291 . . . . . . . . 9 (𝑥𝐽 → (∀𝑏𝐽 (𝑎𝑏𝑦𝑏) → (𝑎𝑥𝑦𝑥)))
2217, 21syl 17 . . . . . . . 8 ((((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) ∧ 𝑎 𝐽) → (∀𝑏𝐽 (𝑎𝑏𝑦𝑏) → (𝑎𝑥𝑦𝑥)))
23223impia 1258 . . . . . . 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 3663 . . . . 5 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ⊆ 𝑥)
26 nrmsep3 21072 . . . . 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 1325 . . . 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 2961 . . . . . . . 8 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → ∀𝑏𝐽 (𝑦𝑏𝑦𝑏))
30 elequ1 1994 . . . . . . . . . . 11 (𝑎 = 𝑦 → (𝑎𝑏𝑦𝑏))
3130bibi1d 333 . . . . . . . . . 10 (𝑎 = 𝑦 → ((𝑎𝑏𝑦𝑏) ↔ (𝑦𝑏𝑦𝑏)))
3231ralbidv 2980 . . . . . . . . 9 (𝑎 = 𝑦 → (∀𝑏𝐽 (𝑎𝑏𝑦𝑏) ↔ ∀𝑏𝐽 (𝑦𝑏𝑦𝑏)))
3332elrab 3347 . . . . . . . 8 (𝑦 ∈ {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)} ↔ (𝑦 𝐽 ∧ ∀𝑏𝐽 (𝑦𝑏𝑦𝑏)))
3412, 29, 33sylanbrc 697 . . . . . . 7 (((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 ∈ {𝑎 𝐽 ∣ ∀𝑏𝐽 (𝑎𝑏𝑦𝑏)})
35 ssel 3578 . . . . . . 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 3010 . . . 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 2965 . 2 ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → ∀𝑥𝐽𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
41 isreg 21049 . 2 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
422, 40, 41sylanbrc 697 1 ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Reg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wcel 1987  wral 2907  wrex 2908  {crab 2911  wss 3556   cuni 4404  cmpt 4675  cfv 5849  Topctop 20620  TopOnctopon 20637  Clsdccld 20733  clsccl 20735  Frect1 21024  Regcreg 21026  Nrmcnrm 21027  KQckq 21409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-map 7807  df-qtop 16091  df-top 20621  df-topon 20638  df-cld 20736  df-cn 20944  df-t1 21031  df-reg 21033  df-nrm 21034  df-kq 21410
This theorem is referenced by:  nrmreg  21540
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