Theorem restclssep 48585

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 4230	. . 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 4230	. . . . 5 ⊢ (𝑆 ∩ 𝑇) = (𝑇 ∩ 𝑆)
8		restclsseplem.6	. . . . 5 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
9	7, 8	eqtr3id 2794	. . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑆) = ∅)
10		restcls2.5	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))
11	2, 3, 4, 5, 10	restcls2lem 48582	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑌)
12	2, 3, 4, 5, 6, 9, 11	restclsseplem 48584	. . 3 ⊢ (𝜑 → (((cls‘𝐽)‘𝑇) ∩ 𝑆) = ∅)
13	1, 12	eqtr3id 2794	. 2 ⊢ (𝜑 → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
14	2, 3, 4, 5, 6	restcls2lem 48582	. . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑌)
15	2, 3, 4, 5, 10, 8, 14	restclsseplem 48584	. 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 1537   ∈ wcel 2108   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ∪ cuni 4931  ‘cfv 6568  (class class class)co 7443   ↾_t crest 17474  Topctop 22912  Clsdccld 23037  clsccl 23039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7764

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5650  df-we 5652  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-ord 6393  df-on 6394  df-lim 6395  df-suc 6396  df-iota 6520  df-fun 6570  df-fn 6571  df-f 6572  df-f1 6573  df-fo 6574  df-f1o 6575  df-fv 6576  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7898  df-1st 8024  df-2nd 8025  df-en 8998  df-fin 9001  df-fi 9474  df-rest 17476  df-topgen 17497  df-top 22913  df-topon 22930  df-bases 22966  df-cld 23040  df-cls 23042

This theorem is referenced by:  iscnrm3l  48621