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Theorem conncompclo 23160
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 = βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}
Assertion
Ref Expression
conncompclo ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)) ∧ 𝐴 ∈ 𝑇) β†’ 𝑆 βŠ† 𝑇)
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐽   π‘₯,𝑋
Allowed substitution hints:   𝑆(π‘₯)   𝑇(π‘₯)
Proof of Theorem conncompclo
StepHypRef Expression
1 eqid 2731 . 2 βˆͺ 𝐽 = βˆͺ 𝐽
2 simp1 1135 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)) ∧ 𝐴 ∈ 𝑇) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
3 simp2 1136 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)) ∧ 𝐴 ∈ 𝑇) β†’ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)))
43elin1d 4199 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)) ∧ 𝐴 ∈ 𝑇) β†’ 𝑇 ∈ 𝐽)
5 toponss 22650 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ 𝐽) β†’ 𝑇 βŠ† 𝑋)
62, 4, 5syl2anc 583 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)) ∧ 𝐴 ∈ 𝑇) β†’ 𝑇 βŠ† 𝑋)
7 simp3 1137 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)) ∧ 𝐴 ∈ 𝑇) β†’ 𝐴 ∈ 𝑇)
86, 7sseldd 3984 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)) ∧ 𝐴 ∈ 𝑇) β†’ 𝐴 ∈ 𝑋)
9 conncomp.2 . . . . 5 𝑆 = βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}
109conncompcld 23159 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 ∈ (Clsdβ€˜π½))
112, 8, 10syl2anc 583 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)) ∧ 𝐴 ∈ 𝑇) β†’ 𝑆 ∈ (Clsdβ€˜π½))
121cldss 22754 . . 3 (𝑆 ∈ (Clsdβ€˜π½) β†’ 𝑆 βŠ† βˆͺ 𝐽)
1311, 12syl 17 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)) ∧ 𝐴 ∈ 𝑇) β†’ 𝑆 βŠ† βˆͺ 𝐽)
149conncompconn 23157 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt 𝑆) ∈ Conn)
152, 8, 14syl2anc 583 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)) ∧ 𝐴 ∈ 𝑇) β†’ (𝐽 β†Ύt 𝑆) ∈ Conn)
169conncompid 23156 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑆)
172, 8, 16syl2anc 583 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)) ∧ 𝐴 ∈ 𝑇) β†’ 𝐴 ∈ 𝑆)
18 inelcm 4465 . . 3 ((𝐴 ∈ 𝑇 ∧ 𝐴 ∈ 𝑆) β†’ (𝑇 ∩ 𝑆) β‰  βˆ…)
197, 17, 18syl2anc 583 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)) ∧ 𝐴 ∈ 𝑇) β†’ (𝑇 ∩ 𝑆) β‰  βˆ…)
203elin2d 4200 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)) ∧ 𝐴 ∈ 𝑇) β†’ 𝑇 ∈ (Clsdβ€˜π½))
211, 13, 15, 4, 19, 20connsubclo 23149 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑇 ∈ (𝐽 ∩ (Clsdβ€˜π½)) ∧ 𝐴 ∈ 𝑇) β†’ 𝑆 βŠ† 𝑇)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  {crab 3431   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βˆͺ cuni 4909  β€˜cfv 6544  (class class class)co 7412   β†Ύt crest 17371  TopOnctopon 22633  Clsdccld 22741  Conncconn 23136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-en 8943  df-fin 8946  df-fi 9409  df-rest 17373  df-topgen 17394  df-top 22617  df-topon 22634  df-bases 22670  df-cld 22744  df-ntr 22745  df-cls 22746  df-conn 23137
This theorem is referenced by:  tgpconncompss  23839
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