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Theorem restclssep 48786
Description: Two disjoint closed sets in a subspace topology are separated in the original topology. (Contributed by Zhi Wang, 2-Sep-2024.)
Hypotheses
Ref Expression
restcls2.1 (𝜑𝐽 ∈ Top)
restcls2.2 (𝜑𝑋 = 𝐽)
restcls2.3 (𝜑𝑌𝑋)
restcls2.4 (𝜑𝐾 = (𝐽t 𝑌))
restcls2.5 (𝜑𝑆 ∈ (Clsd‘𝐾))
restclsseplem.6 (𝜑 → (𝑆𝑇) = ∅)
restclssep.7 (𝜑𝑇 ∈ (Clsd‘𝐾))
Assertion
Ref Expression
restclssep (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))

Proof of Theorem restclssep
StepHypRef Expression
1 incom 4208 . . 3 (((cls‘𝐽)‘𝑇) ∩ 𝑆) = (𝑆 ∩ ((cls‘𝐽)‘𝑇))
2 restcls2.1 . . . 4 (𝜑𝐽 ∈ Top)
3 restcls2.2 . . . 4 (𝜑𝑋 = 𝐽)
4 restcls2.3 . . . 4 (𝜑𝑌𝑋)
5 restcls2.4 . . . 4 (𝜑𝐾 = (𝐽t 𝑌))
6 restclssep.7 . . . 4 (𝜑𝑇 ∈ (Clsd‘𝐾))
7 incom 4208 . . . . 5 (𝑆𝑇) = (𝑇𝑆)
8 restclsseplem.6 . . . . 5 (𝜑 → (𝑆𝑇) = ∅)
97, 8eqtr3id 2790 . . . 4 (𝜑 → (𝑇𝑆) = ∅)
10 restcls2.5 . . . . 5 (𝜑𝑆 ∈ (Clsd‘𝐾))
112, 3, 4, 5, 10restcls2lem 48783 . . . 4 (𝜑𝑆𝑌)
122, 3, 4, 5, 6, 9, 11restclsseplem 48785 . . 3 (𝜑 → (((cls‘𝐽)‘𝑇) ∩ 𝑆) = ∅)
131, 12eqtr3id 2790 . 2 (𝜑 → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
142, 3, 4, 5, 6restcls2lem 48783 . . 3 (𝜑𝑇𝑌)
152, 3, 4, 5, 10, 8, 14restclsseplem 48785 . 2 (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
1613, 15jca 511 1 (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cin 3949  wss 3950  c0 4332   cuni 4905  cfv 6559  (class class class)co 7429  t crest 17461  Topctop 22889  Clsdccld 23014  clsccl 23016
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 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-int 4945  df-iun 4991  df-iin 4992  df-br 5142  df-opab 5204  df-mpt 5224  df-tr 5258  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-ord 6385  df-on 6386  df-lim 6387  df-suc 6388  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-om 7884  df-1st 8010  df-2nd 8011  df-en 8982  df-fin 8985  df-fi 9447  df-rest 17463  df-topgen 17484  df-top 22890  df-topon 22907  df-bases 22943  df-cld 23017  df-cls 23019
This theorem is referenced by:  iscnrm3l  48821
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