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Theorem isnrm2 21442
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 21420 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2 nrmsep2 21440 . . . . . 6 ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐𝑑) = ∅)) → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))
323exp2 1463 . . . . 5 (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))))
43impd 398 . . . 4 (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
54ralrimivv 3117 . . 3 (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))
61, 5jca 507 . 2 (𝐽 ∈ Nrm → (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
7 simpl 474 . . 3 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))) → 𝐽 ∈ Top)
8 eqid 2765 . . . . . . . . . . 11 𝐽 = 𝐽
98opncld 21117 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
109adantr 472 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
11 ineq2 3970 . . . . . . . . . . . 12 (𝑑 = ( 𝐽𝑥) → (𝑐𝑑) = (𝑐 ∩ ( 𝐽𝑥)))
1211eqeq1d 2767 . . . . . . . . . . 11 (𝑑 = ( 𝐽𝑥) → ((𝑐𝑑) = ∅ ↔ (𝑐 ∩ ( 𝐽𝑥)) = ∅))
13 ineq2 3970 . . . . . . . . . . . . . 14 (𝑑 = ( 𝐽𝑥) → (((cls‘𝐽)‘𝑜) ∩ 𝑑) = (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)))
1413eqeq1d 2767 . . . . . . . . . . . . 13 (𝑑 = ( 𝐽𝑥) → ((((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅ ↔ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅))
1514anbi2d 622 . . . . . . . . . . . 12 (𝑑 = ( 𝐽𝑥) → ((𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅) ↔ (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)))
1615rexbidv 3199 . . . . . . . . . . 11 (𝑑 = ( 𝐽𝑥) → (∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅) ↔ ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)))
1712, 16imbi12d 335 . . . . . . . . . 10 (𝑑 = ( 𝐽𝑥) → (((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) ↔ ((𝑐 ∩ ( 𝐽𝑥)) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅))))
1817rspcv 3457 . . . . . . . . 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 4112 . . . . . . . . . 10 ((𝑐 𝐽) ⊆ 𝑥 ↔ (𝑐 ∩ ( 𝐽𝑥)) = ∅)
218cldss 21113 . . . . . . . . . . . . 13 (𝑐 ∈ (Clsd‘𝐽) → 𝑐 𝐽)
2221adantl 473 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝑐 𝐽)
23 df-ss 3746 . . . . . . . . . . . 12 (𝑐 𝐽 ↔ (𝑐 𝐽) = 𝑐)
2422, 23sylib 209 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝑐 𝐽) = 𝑐)
2524sseq1d 3792 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 𝐽) ⊆ 𝑥𝑐𝑥))
2620, 25syl5bbr 276 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ ( 𝐽𝑥)) = ∅ ↔ 𝑐𝑥))
27 inssdif0 4112 . . . . . . . . . . . 12 ((((cls‘𝐽)‘𝑜) ∩ 𝐽) ⊆ 𝑥 ↔ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)
28 simpll 783 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
29 elssuni 4625 . . . . . . . . . . . . . . 15 (𝑜𝐽𝑜 𝐽)
308clsss3 21143 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑜 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝐽)
3128, 29, 30syl2an 589 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝐽)
32 df-ss 3746 . . . . . . . . . . . . . 14 (((cls‘𝐽)‘𝑜) ⊆ 𝐽 ↔ (((cls‘𝐽)‘𝑜) ∩ 𝐽) = ((cls‘𝐽)‘𝑜))
3331, 32sylib 209 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → (((cls‘𝐽)‘𝑜) ∩ 𝐽) = ((cls‘𝐽)‘𝑜))
3433sseq1d 3792 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((((cls‘𝐽)‘𝑜) ∩ 𝐽) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
3527, 34syl5bbr 276 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅ ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
3635anbi2d 622 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅) ↔ (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
3736rexbidva 3196 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅) ↔ ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
3826, 37imbi12d 335 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (((𝑐 ∩ ( 𝐽𝑥)) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)) ↔ (𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
3919, 38sylibd 230 . . . . . . 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 3109 . . . . . 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 3958 . . . . . . . . . 10 (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥))
42 selpw 4322 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 𝑥𝑐𝑥)
4342anbi2i 616 . . . . . . . . . 10 ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥))
4441, 43bitri 266 . . . . . . . . 9 (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥))
4544imbi1i 340 . . . . . . . 8 ((𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
46 impexp 441 . . . . . . . 8 (((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ (𝑐 ∈ (Clsd‘𝐽) → (𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
4745, 46bitri 266 . . . . . . 7 ((𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ (𝑐 ∈ (Clsd‘𝐽) → (𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
4847ralbii2 3125 . . . . . 6 (∀𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥) ↔ ∀𝑐 ∈ (Clsd‘𝐽)(𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
4940, 48syl6ibr 243 . . . . 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 3116 . . . 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 395 . . 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 21419 . . 3 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
537, 51, 52sylanbrc 578 . 2 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))) → 𝐽 ∈ Nrm)
546, 53impbii 200 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 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  cdif 3729  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315   cuni 4594  cfv 6068  Topctop 20977  Clsdccld 21100  clsccl 21102  Nrmcnrm 21394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-top 20978  df-cld 21103  df-cls 21105  df-nrm 21401
This theorem is referenced by:  isnrm3  21443
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