Theorem restclssep 47934

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 4201	. . 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 4201	. . . . 5 ⊢ (𝑆 ∩ 𝑇) = (𝑇 ∩ 𝑆)
8		restclsseplem.6	. . . . 5 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
9	7, 8	eqtr3id 2782	. . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑆) = ∅)
10		restcls2.5	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))
11	2, 3, 4, 5, 10	restcls2lem 47931	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑌)
12	2, 3, 4, 5, 6, 9, 11	restclsseplem 47933	. . 3 ⊢ (𝜑 → (((cls‘𝐽)‘𝑇) ∩ 𝑆) = ∅)
13	1, 12	eqtr3id 2782	. 2 ⊢ (𝜑 → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
14	2, 3, 4, 5, 6	restcls2lem 47931	. . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑌)
15	2, 3, 4, 5, 10, 8, 14	restclsseplem 47933	. 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 1534   ∈ wcel 2099   ∩ cin 3946   ⊆ wss 3947  ∅c0 4323  ∪ cuni 4908  ‘cfv 6548  (class class class)co 7420   ↾_t crest 17402  Topctop 22808  Clsdccld 22933  clsccl 22935

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 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-en 8965  df-fin 8968  df-fi 9435  df-rest 17404  df-topgen 17425  df-top 22809  df-topon 22826  df-bases 22862  df-cld 22936  df-cls 22938

This theorem is referenced by:  iscnrm3l  47970

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