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	⊢ (𝜑 → 𝐽 ∈ Top)
restcls2.2	⊢ (𝜑 → 𝑋 = ∪ 𝐽)
restcls2.3	⊢ (𝜑 → 𝑌 ⊆ 𝑋)
restcls2.4	⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))
restcls2.5	⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))
restclsseplem.6	⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
restclssep.7	⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐾))

Assertion

Ref	Expression
restclssep	⊢ (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))

Proof of Theorem restclssep

Step	Hyp	Ref	Expression
1		incom 4208	. . 3 ⊢ (((cls‘𝐽)‘𝑇) ∩ 𝑆) = (𝑆 ∩ ((cls‘𝐽)‘𝑇))
2		restcls2.1	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Top)
3		restcls2.2	. . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
4		restcls2.3	. . . 4 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
5		restcls2.4	. . . 4 ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))
6		restclssep.7	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐾))
7		incom 4208	. . . . 5 ⊢ (𝑆 ∩ 𝑇) = (𝑇 ∩ 𝑆)
8		restclsseplem.6	. . . . 5 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
9	7, 8	eqtr3id 2790	. . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑆) = ∅)
10		restcls2.5	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))
11	2, 3, 4, 5, 10	restcls2lem 48783	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑌)
12	2, 3, 4, 5, 6, 9, 11	restclsseplem 48785	. . 3 ⊢ (𝜑 → (((cls‘𝐽)‘𝑇) ∩ 𝑆) = ∅)
13	1, 12	eqtr3id 2790	. 2 ⊢ (𝜑 → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
14	2, 3, 4, 5, 6	restcls2lem 48783	. . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑌)
15	2, 3, 4, 5, 10, 8, 14	restclsseplem 48785	. 2 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
16	13, 15	jca 511	1 ⊢ (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))

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