Theorem conncompclo 23400

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 2736	. 2 ⊢ ∪ 𝐽 = ∪ 𝐽
2		simp1 1137	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝐽 ∈ (TopOn‘𝑋))
3		simp2 1138	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)))
4	3	elin1d 4144	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑇 ∈ 𝐽)
5		toponss 22892	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ 𝐽) → 𝑇 ⊆ 𝑋)
6	2, 4, 5	syl2anc 585	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ 𝑋)
7		simp3 1139	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑇)
8	6, 7	sseldd 3922	. . . 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 23399	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝑆 ∈ (Clsd‘𝐽))
11	2, 8, 10	syl2anc 585	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑆 ∈ (Clsd‘𝐽))
12	1	cldss 22994	. . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽)
13	11, 12	syl 17	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑆 ⊆ ∪ 𝐽)
14	9	conncompconn 23397	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t 𝑆) ∈ Conn)
15	2, 8, 14	syl2anc 585	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → (𝐽 ↾t 𝑆) ∈ Conn)
16	9	conncompid 23396	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑆)
17	2, 8, 16	syl2anc 585	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑆)
18		inelcm 4405	. . 3 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑆) → (𝑇 ∩ 𝑆) ≠ ∅)
19	7, 17, 18	syl2anc 585	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → (𝑇 ∩ 𝑆) ≠ ∅)
20	3	elin2d 4145	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑇 ∈ (Clsd‘𝐽))
21	1, 13, 15, 4, 19, 20	connsubclo 23389	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑇 ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝐴 ∈ 𝑇) → 𝑆 ⊆ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  {crab 3389   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  ∪ cuni 4850  ‘cfv 6498  (class class class)co 7367   ↾_t crest 17383  TopOnctopon 22875  Clsdccld 22981  Conncconn 23376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-en 8894  df-fin 8897  df-fi 9324  df-rest 17385  df-topgen 17406  df-top 22859  df-topon 22876  df-bases 22911  df-cld 22984  df-ntr 22985  df-cls 22986  df-conn 23377

This theorem is referenced by:  tgpconncompss  24079