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Theorem isnrm2 23382
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 23360 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2 nrmsep2 23380 . . . . . 6 ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐𝑑) = ∅)) → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))
323exp2 1353 . . . . 5 (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))))
43impd 410 . . . 4 (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
54ralrimivv 3198 . . 3 (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))
61, 5jca 511 . 2 (𝐽 ∈ Nrm → (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
7 simpl 482 . . 3 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))) → 𝐽 ∈ Top)
8 eqid 2735 . . . . . . . . . . 11 𝐽 = 𝐽
98opncld 23057 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
109adantr 480 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
11 ineq2 4222 . . . . . . . . . . . 12 (𝑑 = ( 𝐽𝑥) → (𝑐𝑑) = (𝑐 ∩ ( 𝐽𝑥)))
1211eqeq1d 2737 . . . . . . . . . . 11 (𝑑 = ( 𝐽𝑥) → ((𝑐𝑑) = ∅ ↔ (𝑐 ∩ ( 𝐽𝑥)) = ∅))
13 ineq2 4222 . . . . . . . . . . . . . 14 (𝑑 = ( 𝐽𝑥) → (((cls‘𝐽)‘𝑜) ∩ 𝑑) = (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)))
1413eqeq1d 2737 . . . . . . . . . . . . 13 (𝑑 = ( 𝐽𝑥) → ((((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅ ↔ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅))
1514anbi2d 630 . . . . . . . . . . . 12 (𝑑 = ( 𝐽𝑥) → ((𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅) ↔ (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)))
1615rexbidv 3177 . . . . . . . . . . 11 (𝑑 = ( 𝐽𝑥) → (∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅) ↔ ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)))
1712, 16imbi12d 344 . . . . . . . . . 10 (𝑑 = ( 𝐽𝑥) → (((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) ↔ ((𝑐 ∩ ( 𝐽𝑥)) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅))))
1817rspcv 3618 . . . . . . . . 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 4380 . . . . . . . . . 10 ((𝑐 𝐽) ⊆ 𝑥 ↔ (𝑐 ∩ ( 𝐽𝑥)) = ∅)
218cldss 23053 . . . . . . . . . . . . 13 (𝑐 ∈ (Clsd‘𝐽) → 𝑐 𝐽)
2221adantl 481 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝑐 𝐽)
23 dfss2 3981 . . . . . . . . . . . 12 (𝑐 𝐽 ↔ (𝑐 𝐽) = 𝑐)
2422, 23sylib 218 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝑐 𝐽) = 𝑐)
2524sseq1d 4027 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 𝐽) ⊆ 𝑥𝑐𝑥))
2620, 25bitr3id 285 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ ( 𝐽𝑥)) = ∅ ↔ 𝑐𝑥))
27 inssdif0 4380 . . . . . . . . . . . 12 ((((cls‘𝐽)‘𝑜) ∩ 𝐽) ⊆ 𝑥 ↔ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)
28 simpll 767 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
29 elssuni 4942 . . . . . . . . . . . . . . 15 (𝑜𝐽𝑜 𝐽)
308clsss3 23083 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑜 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝐽)
3128, 29, 30syl2an 596 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝐽)
32 dfss2 3981 . . . . . . . . . . . . . 14 (((cls‘𝐽)‘𝑜) ⊆ 𝐽 ↔ (((cls‘𝐽)‘𝑜) ∩ 𝐽) = ((cls‘𝐽)‘𝑜))
3331, 32sylib 218 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → (((cls‘𝐽)‘𝑜) ∩ 𝐽) = ((cls‘𝐽)‘𝑜))
3433sseq1d 4027 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((((cls‘𝐽)‘𝑜) ∩ 𝐽) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
3527, 34bitr3id 285 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅ ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
3635anbi2d 630 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅) ↔ (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
3736rexbidva 3175 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅) ↔ ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
3826, 37imbi12d 344 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (((𝑐 ∩ ( 𝐽𝑥)) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)) ↔ (𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
3919, 38sylibd 239 . . . . . . 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 3165 . . . . . 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 3979 . . . . . . . . . 10 (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥))
42 velpw 4610 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 𝑥𝑐𝑥)
4342anbi2i 623 . . . . . . . . . 10 ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥))
4441, 43bitri 275 . . . . . . . . 9 (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥))
4544imbi1i 349 . . . . . . . 8 ((𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
46 impexp 450 . . . . . . . 8 (((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ (𝑐 ∈ (Clsd‘𝐽) → (𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
4745, 46bitri 275 . . . . . . 7 ((𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ (𝑐 ∈ (Clsd‘𝐽) → (𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
4847ralbii2 3087 . . . . . 6 (∀𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥) ↔ ∀𝑐 ∈ (Clsd‘𝐽)(𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
4940, 48imbitrrdi 252 . . . . 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 3152 . . . 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 406 . . 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 23359 . . 3 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
537, 51, 52sylanbrc 583 . 2 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))) → 𝐽 ∈ Nrm)
546, 53impbii 209 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 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cdif 3960  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912  cfv 6563  Topctop 22915  Clsdccld 23040  clsccl 23042  Nrmcnrm 23334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-cld 23043  df-cls 23045  df-nrm 23341
This theorem is referenced by:  isnrm3  23383
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