Theorem restclssep 46097

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 4131	. . 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 4131	. . . . 5 ⊢ (𝑆 ∩ 𝑇) = (𝑇 ∩ 𝑆)
8		restclsseplem.6	. . . . 5 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
9	7, 8	eqtr3id 2793	. . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑆) = ∅)
10		restcls2.5	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾))
11	2, 3, 4, 5, 10	restcls2lem 46094	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑌)
12	2, 3, 4, 5, 6, 9, 11	restclsseplem 46096	. . 3 ⊢ (𝜑 → (((cls‘𝐽)‘𝑇) ∩ 𝑆) = ∅)
13	1, 12	eqtr3id 2793	. 2 ⊢ (𝜑 → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
14	2, 3, 4, 5, 6	restcls2lem 46094	. . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑌)
15	2, 3, 4, 5, 10, 8, 14	restclsseplem 46096	. 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 1539   ∈ wcel 2108   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836  ‘cfv 6418  (class class class)co 7255   ↾_t crest 17048  Topctop 21950  Clsdccld 22075  clsccl 22077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-en 8692  df-fin 8695  df-fi 9100  df-rest 17050  df-topgen 17071  df-top 21951  df-topon 21968  df-bases 22004  df-cld 22078  df-cls 22080

This theorem is referenced by:  iscnrm3l  46133