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Theorem isnrm2 23366
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 23344 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2 nrmsep2 23364 . . . . . 6 ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐𝑑) = ∅)) → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))
323exp2 1355 . . . . 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 3200 . . 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 2737 . . . . . . . . . . 11 𝐽 = 𝐽
98opncld 23041 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
109adantr 480 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
11 ineq2 4214 . . . . . . . . . . . 12 (𝑑 = ( 𝐽𝑥) → (𝑐𝑑) = (𝑐 ∩ ( 𝐽𝑥)))
1211eqeq1d 2739 . . . . . . . . . . 11 (𝑑 = ( 𝐽𝑥) → ((𝑐𝑑) = ∅ ↔ (𝑐 ∩ ( 𝐽𝑥)) = ∅))
13 ineq2 4214 . . . . . . . . . . . . . 14 (𝑑 = ( 𝐽𝑥) → (((cls‘𝐽)‘𝑜) ∩ 𝑑) = (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)))
1413eqeq1d 2739 . . . . . . . . . . . . 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 3179 . . . . . . . . . . 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 4374 . . . . . . . . . 10 ((𝑐 𝐽) ⊆ 𝑥 ↔ (𝑐 ∩ ( 𝐽𝑥)) = ∅)
218cldss 23037 . . . . . . . . . . . . 13 (𝑐 ∈ (Clsd‘𝐽) → 𝑐 𝐽)
2221adantl 481 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝑐 𝐽)
23 dfss2 3969 . . . . . . . . . . . 12 (𝑐 𝐽 ↔ (𝑐 𝐽) = 𝑐)
2422, 23sylib 218 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝑐 𝐽) = 𝑐)
2524sseq1d 4015 . . . . . . . . . 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 4374 . . . . . . . . . . . 12 ((((cls‘𝐽)‘𝑜) ∩ 𝐽) ⊆ 𝑥 ↔ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)
28 simpll 767 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
29 elssuni 4937 . . . . . . . . . . . . . . 15 (𝑜𝐽𝑜 𝐽)
308clsss3 23067 . . . . . . . . . . . . . . 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 3969 . . . . . . . . . . . . . 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 4015 . . . . . . . . . . . 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 3177 . . . . . . . . 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 3167 . . . . . 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 3967 . . . . . . . . . 10 (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥))
42 velpw 4605 . . . . . . . . . . 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 3089 . . . . . 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 3154 . . . 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 23343 . . 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 1540  wcel 2108  wral 3061  wrex 3070  cdif 3948  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600   cuni 4907  cfv 6561  Topctop 22899  Clsdccld 23024  clsccl 23026  Nrmcnrm 23318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027  df-cls 23029  df-nrm 23325
This theorem is referenced by:  isnrm3  23367
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