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Theorem isnrm2 20972
Description: An alternate characterization of normality. This is the important property in the proof of Urysohn's lemma. (Contributed by Jeff Hankins, 1-Feb-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
isnrm2 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
Distinct variable group:   𝑐,𝑑,𝑜,𝐽

Proof of Theorem isnrm2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nrmtop 20950 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2 nrmsep2 20970 . . . . . 6 ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐𝑑) = ∅)) → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))
323exp2 1277 . . . . 5 (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))))
43impd 446 . . . 4 (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
54ralrimivv 2953 . . 3 (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))
61, 5jca 553 . 2 (𝐽 ∈ Nrm → (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
7 simpl 472 . . 3 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))) → 𝐽 ∈ Top)
8 eqid 2610 . . . . . . . . . . 11 𝐽 = 𝐽
98opncld 20647 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
109adantr 480 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
11 ineq2 3770 . . . . . . . . . . . 12 (𝑑 = ( 𝐽𝑥) → (𝑐𝑑) = (𝑐 ∩ ( 𝐽𝑥)))
1211eqeq1d 2612 . . . . . . . . . . 11 (𝑑 = ( 𝐽𝑥) → ((𝑐𝑑) = ∅ ↔ (𝑐 ∩ ( 𝐽𝑥)) = ∅))
13 ineq2 3770 . . . . . . . . . . . . . 14 (𝑑 = ( 𝐽𝑥) → (((cls‘𝐽)‘𝑜) ∩ 𝑑) = (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)))
1413eqeq1d 2612 . . . . . . . . . . . . 13 (𝑑 = ( 𝐽𝑥) → ((((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅ ↔ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅))
1514anbi2d 736 . . . . . . . . . . . 12 (𝑑 = ( 𝐽𝑥) → ((𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅) ↔ (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)))
1615rexbidv 3034 . . . . . . . . . . 11 (𝑑 = ( 𝐽𝑥) → (∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅) ↔ ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)))
1712, 16imbi12d 333 . . . . . . . . . 10 (𝑑 = ( 𝐽𝑥) → (((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) ↔ ((𝑐 ∩ ( 𝐽𝑥)) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅))))
1817rspcv 3278 . . . . . . . . 9 (( 𝐽𝑥) ∈ (Clsd‘𝐽) → (∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) → ((𝑐 ∩ ( 𝐽𝑥)) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅))))
1910, 18syl 17 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) → ((𝑐 ∩ ( 𝐽𝑥)) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅))))
20 inssdif0 3901 . . . . . . . . . 10 ((𝑐 𝐽) ⊆ 𝑥 ↔ (𝑐 ∩ ( 𝐽𝑥)) = ∅)
218cldss 20643 . . . . . . . . . . . . 13 (𝑐 ∈ (Clsd‘𝐽) → 𝑐 𝐽)
2221adantl 481 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝑐 𝐽)
23 df-ss 3554 . . . . . . . . . . . 12 (𝑐 𝐽 ↔ (𝑐 𝐽) = 𝑐)
2422, 23sylib 207 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝑐 𝐽) = 𝑐)
2524sseq1d 3595 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 𝐽) ⊆ 𝑥𝑐𝑥))
2620, 25syl5bbr 273 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ ( 𝐽𝑥)) = ∅ ↔ 𝑐𝑥))
27 inssdif0 3901 . . . . . . . . . . . 12 ((((cls‘𝐽)‘𝑜) ∩ 𝐽) ⊆ 𝑥 ↔ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)
28 simpll 786 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
29 elssuni 4403 . . . . . . . . . . . . . . 15 (𝑜𝐽𝑜 𝐽)
308clsss3 20673 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑜 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝐽)
3128, 29, 30syl2an 493 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝐽)
32 df-ss 3554 . . . . . . . . . . . . . 14 (((cls‘𝐽)‘𝑜) ⊆ 𝐽 ↔ (((cls‘𝐽)‘𝑜) ∩ 𝐽) = ((cls‘𝐽)‘𝑜))
3331, 32sylib 207 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → (((cls‘𝐽)‘𝑜) ∩ 𝐽) = ((cls‘𝐽)‘𝑜))
3433sseq1d 3595 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((((cls‘𝐽)‘𝑜) ∩ 𝐽) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
3527, 34syl5bbr 273 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅ ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
3635anbi2d 736 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅) ↔ (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
3736rexbidva 3031 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅) ↔ ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
3826, 37imbi12d 333 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (((𝑐 ∩ ( 𝐽𝑥)) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)) ↔ (𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
3919, 38sylibd 228 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) → (𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
4039ralimdva 2945 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) → ∀𝑐 ∈ (Clsd‘𝐽)(𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
41 elin 3758 . . . . . . . . . 10 (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥))
42 selpw 4115 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 𝑥𝑐𝑥)
4342anbi2i 726 . . . . . . . . . 10 ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥))
4441, 43bitri 263 . . . . . . . . 9 (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥))
4544imbi1i 338 . . . . . . . 8 ((𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
46 impexp 461 . . . . . . . 8 (((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ (𝑐 ∈ (Clsd‘𝐽) → (𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
4745, 46bitri 263 . . . . . . 7 ((𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ (𝑐 ∈ (Clsd‘𝐽) → (𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
4847ralbii2 2961 . . . . . 6 (∀𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥) ↔ ∀𝑐 ∈ (Clsd‘𝐽)(𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
4940, 48syl6ibr 241 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝐽) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) → ∀𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
5049ralrimdva 2952 . . . 4 (𝐽 ∈ Top → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) → ∀𝑥𝐽𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
5150imp 444 . . 3 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))) → ∀𝑥𝐽𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
52 isnrm 20949 . . 3 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
537, 51, 52sylanbrc 695 . 2 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))) → 𝐽 ∈ Nrm)
546, 53impbii 198 1 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897  cdif 3537  cin 3539  wss 3540  c0 3874  𝒫 cpw 4108   cuni 4372  cfv 5804  Topctop 20517  Clsdccld 20630  clsccl 20632  Nrmcnrm 20924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-cld 20633  df-cls 20635  df-nrm 20931
This theorem is referenced by:  isnrm3  20973
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