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Theorem isnrm2 21885
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 21863 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2 nrmsep2 21883 . . . . . 6 ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐𝑑) = ∅)) → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))
323exp2 1348 . . . . 5 (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))))
43impd 411 . . . 4 (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
54ralrimivv 3195 . . 3 (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))
61, 5jca 512 . 2 (𝐽 ∈ Nrm → (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
7 simpl 483 . . 3 ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))) → 𝐽 ∈ Top)
8 eqid 2826 . . . . . . . . . . 11 𝐽 = 𝐽
98opncld 21560 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
109adantr 481 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
11 ineq2 4187 . . . . . . . . . . . 12 (𝑑 = ( 𝐽𝑥) → (𝑐𝑑) = (𝑐 ∩ ( 𝐽𝑥)))
1211eqeq1d 2828 . . . . . . . . . . 11 (𝑑 = ( 𝐽𝑥) → ((𝑐𝑑) = ∅ ↔ (𝑐 ∩ ( 𝐽𝑥)) = ∅))
13 ineq2 4187 . . . . . . . . . . . . . 14 (𝑑 = ( 𝐽𝑥) → (((cls‘𝐽)‘𝑜) ∩ 𝑑) = (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)))
1413eqeq1d 2828 . . . . . . . . . . . . 13 (𝑑 = ( 𝐽𝑥) → ((((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅ ↔ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅))
1514anbi2d 628 . . . . . . . . . . . 12 (𝑑 = ( 𝐽𝑥) → ((𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅) ↔ (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)))
1615rexbidv 3302 . . . . . . . . . . 11 (𝑑 = ( 𝐽𝑥) → (∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅) ↔ ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)))
1712, 16imbi12d 346 . . . . . . . . . 10 (𝑑 = ( 𝐽𝑥) → (((𝑐𝑑) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) ↔ ((𝑐 ∩ ( 𝐽𝑥)) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅))))
1817rspcv 3622 . . . . . . . . 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 4333 . . . . . . . . . 10 ((𝑐 𝐽) ⊆ 𝑥 ↔ (𝑐 ∩ ( 𝐽𝑥)) = ∅)
218cldss 21556 . . . . . . . . . . . . 13 (𝑐 ∈ (Clsd‘𝐽) → 𝑐 𝐽)
2221adantl 482 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝑐 𝐽)
23 df-ss 3956 . . . . . . . . . . . 12 (𝑐 𝐽 ↔ (𝑐 𝐽) = 𝑐)
2422, 23sylib 219 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝑐 𝐽) = 𝑐)
2524sseq1d 4002 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 𝐽) ⊆ 𝑥𝑐𝑥))
2620, 25syl5bbr 286 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ ( 𝐽𝑥)) = ∅ ↔ 𝑐𝑥))
27 inssdif0 4333 . . . . . . . . . . . 12 ((((cls‘𝐽)‘𝑜) ∩ 𝐽) ⊆ 𝑥 ↔ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)
28 simpll 763 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
29 elssuni 4866 . . . . . . . . . . . . . . 15 (𝑜𝐽𝑜 𝐽)
308clsss3 21586 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑜 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝐽)
3128, 29, 30syl2an 595 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((cls‘𝐽)‘𝑜) ⊆ 𝐽)
32 df-ss 3956 . . . . . . . . . . . . . 14 (((cls‘𝐽)‘𝑜) ⊆ 𝐽 ↔ (((cls‘𝐽)‘𝑜) ∩ 𝐽) = ((cls‘𝐽)‘𝑜))
3331, 32sylib 219 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → (((cls‘𝐽)‘𝑜) ∩ 𝐽) = ((cls‘𝐽)‘𝑜))
3433sseq1d 4002 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((((cls‘𝐽)‘𝑜) ∩ 𝐽) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
3527, 34syl5bbr 286 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅ ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
3635anbi2d 628 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜𝐽) → ((𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅) ↔ (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
3736rexbidva 3301 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅) ↔ ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
3826, 37imbi12d 346 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑥𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (((𝑐 ∩ ( 𝐽𝑥)) = ∅ → ∃𝑜𝐽 (𝑐𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ ( 𝐽𝑥)) = ∅)) ↔ (𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
3919, 38sylibd 240 . . . . . . 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 3182 . . . . . 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 4173 . . . . . . . . . 10 (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥))
42 velpw 4550 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 𝑥𝑐𝑥)
4342anbi2i 622 . . . . . . . . . 10 ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥))
4441, 43bitri 276 . . . . . . . . 9 (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥))
4544imbi1i 351 . . . . . . . 8 ((𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
46 impexp 451 . . . . . . . 8 (((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ (𝑐 ∈ (Clsd‘𝐽) → (𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
4745, 46bitri 276 . . . . . . 7 ((𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ (𝑐 ∈ (Clsd‘𝐽) → (𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
4847ralbii2 3168 . . . . . 6 (∀𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥) ↔ ∀𝑐 ∈ (Clsd‘𝐽)(𝑐𝑥 → ∃𝑜𝐽 (𝑐𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
4940, 48syl6ibr 253 . . . . 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 3194 . . . 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 407 . . 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 21862 . . 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 210 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 207  wa 396   = wceq 1530  wcel 2107  wral 3143  wrex 3144  cdif 3937  cin 3939  wss 3940  c0 4295  𝒫 cpw 4542   cuni 4837  cfv 6352  Topctop 21420  Clsdccld 21543  clsccl 21545  Nrmcnrm 21837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-top 21421  df-cld 21546  df-cls 21548  df-nrm 21844
This theorem is referenced by:  isnrm3  21886
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