Theorem restclssep 49613

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 4170	. . 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 4170	. . . . 5 ⊢ (𝑆 ∩ 𝑇) = (𝑇 ∩ 𝑆)
8		restclsseplem.6	. . . . 5 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
9	7, 8	eqtr3id 2818	. . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑆) = ∅)
10		restcls2.5	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))
11	2, 3, 4, 5, 10	restcls2lem 49610	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑌)
12	2, 3, 4, 5, 6, 9, 11	restclsseplem 49612	. . 3 ⊢ (𝜑 → (((cls‘𝐽)‘𝑇) ∩ 𝑆) = ∅)
13	1, 12	eqtr3id 2818	. 2 ⊢ (𝜑 → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
14	2, 3, 4, 5, 6	restcls2lem 49610	. . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑌)
15	2, 3, 4, 5, 10, 8, 14	restclsseplem 49612	. 2 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
16	13, 15	jca 520	1 ⊢ (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  ∪ cuni 4876  ‘cfv 6537  (class class class)co 7411   ↾_t crest 17473  Topctop 23019  Clsdccld 23142  clsccl 23144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-en 8944  df-fin 8947  df-fi 9371  df-rest 17475  df-topgen 17496  df-top 23020  df-topon 23037  df-bases 23072  df-cld 23145  df-cls 23147

This theorem is referenced by:  iscnrm3l  49648