Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  restclssep Structured version   Visualization version   GIF version

Theorem restclssep 48633
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 4217 . . 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 4217 . . . . 5 (𝑆𝑇) = (𝑇𝑆)
8 restclsseplem.6 . . . . 5 (𝜑 → (𝑆𝑇) = ∅)
97, 8eqtr3id 2787 . . . 4 (𝜑 → (𝑇𝑆) = ∅)
10 restcls2.5 . . . . 5 (𝜑𝑆 ∈ (Clsd‘𝐾))
112, 3, 4, 5, 10restcls2lem 48630 . . . 4 (𝜑𝑆𝑌)
122, 3, 4, 5, 6, 9, 11restclsseplem 48632 . . 3 (𝜑 → (((cls‘𝐽)‘𝑇) ∩ 𝑆) = ∅)
131, 12eqtr3id 2787 . 2 (𝜑 → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
142, 3, 4, 5, 6restcls2lem 48630 . . 3 (𝜑𝑇𝑌)
152, 3, 4, 5, 10, 8, 14restclsseplem 48632 . 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 1535  wcel 2104  cin 3962  wss 3963  c0 4339   cuni 4914  cfv 6558  (class class class)co 7425  t crest 17456  Topctop 22896  Clsdccld 23021  clsccl 23023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5366  ax-pr 5430  ax-un 7747
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3377  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-int 4954  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  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 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-om 7881  df-1st 8007  df-2nd 8008  df-en 8979  df-fin 8982  df-fi 9442  df-rest 17458  df-topgen 17479  df-top 22897  df-topon 22914  df-bases 22950  df-cld 23024  df-cls 23026
This theorem is referenced by:  iscnrm3l  48669
  Copyright terms: Public domain W3C validator