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Theorem restclssep 46703
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 4160 . . 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 4160 . . . . 5 (𝑆𝑇) = (𝑇𝑆)
8 restclsseplem.6 . . . . 5 (𝜑 → (𝑆𝑇) = ∅)
97, 8eqtr3id 2792 . . . 4 (𝜑 → (𝑇𝑆) = ∅)
10 restcls2.5 . . . . 5 (𝜑𝑆 ∈ (Clsd‘𝐾))
112, 3, 4, 5, 10restcls2lem 46700 . . . 4 (𝜑𝑆𝑌)
122, 3, 4, 5, 6, 9, 11restclsseplem 46702 . . 3 (𝜑 → (((cls‘𝐽)‘𝑇) ∩ 𝑆) = ∅)
131, 12eqtr3id 2792 . 2 (𝜑 → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
142, 3, 4, 5, 6restcls2lem 46700 . . 3 (𝜑𝑇𝑌)
152, 3, 4, 5, 10, 8, 14restclsseplem 46702 . 2 (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
1613, 15jca 513 1 (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cin 3908  wss 3909  c0 4281   cuni 4864  cfv 6492  (class class class)co 7350  t crest 17237  Topctop 22165  Clsdccld 22290  clsccl 22292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-iun 4955  df-iin 4956  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7353  df-oprab 7354  df-mpo 7355  df-om 7794  df-1st 7912  df-2nd 7913  df-en 8818  df-fin 8821  df-fi 9281  df-rest 17239  df-topgen 17260  df-top 22166  df-topon 22183  df-bases 22219  df-cld 22293  df-cls 22295
This theorem is referenced by:  iscnrm3l  46739
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