Theorem conncompclo 23557

Description: The connected component containing 𝐴 is a subset of any clopen set containing 𝐴. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypothesis

Ref	Expression
conncomp.2	⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}

Assertion

Ref	Expression
conncompclo	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑆 ⊆ 𝑇)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋

Allowed substitution hints:   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem conncompclo

Step	Hyp	Ref	Expression
1		eqid 2769	. 2 ⊢ ∪ 𝐽 = ∪ 𝐽
2		simp1 1152	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝐽 ∈ (TopOn‘𝑋))
3		simp2 1153	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)))
4	3	elin1d 4165	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑇 ∈ 𝐽)
5		toponss 23049	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ 𝐽) → 𝑇 ⊆ 𝑋)
6	2, 4, 5	syl2anc 595	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ 𝑋)
7		simp3 1154	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑇)
8	6, 7	sseldd 3946	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑋)
9		conncomp.2	. . . . 5 ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}
10	9	conncompcld 23556	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝑆 ∈ (Clsd‘𝐽))
11	2, 8, 10	syl2anc 595	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑆 ∈ (Clsd‘𝐽))
12	1	cldss 23151	. . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽)
13	11, 12	syl 18	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑆 ⊆ ∪ 𝐽)
14	9	conncompconn 23554	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t 𝑆) ∈ Conn)
15	2, 8, 14	syl2anc 595	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → (𝐽 ↾t 𝑆) ∈ Conn)
16	9	conncompid 23553	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑆)
17	2, 8, 16	syl2anc 595	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑆)
18		inelcm 4428	. . 3 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑆) → (𝑇 ∩ 𝑆) ≠ ∅)
19	7, 17, 18	syl2anc 595	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → (𝑇 ∩ 𝑆) ≠ ∅)
20	3	elin2d 4166	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑇 ∈ (Clsd‘𝐽))
21	1, 13, 15, 4, 19, 20	connsubclo 23546	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑆 ⊆ 𝑇)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  {crab 3423   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4564  ∪ cuni 4873  ‘cfv 6534  (class class class)co 7408   ↾_t crest 17469  TopOnctopon 23032  Clsdccld 23138  Conncconn 23533

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 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

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 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-en 8940  df-fin 8943  df-fi 9367  df-rest 17471  df-topgen 17492  df-top 23016  df-topon 23033  df-bases 23068  df-cld 23141  df-ntr 23142  df-cls 23143  df-conn 23534

This theorem is referenced by:  tgpconncompss  24236