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Theorem concompclo 17161
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
concomp.2  |-  S  = 
U. { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }
Assertion
Ref Expression
concompclo  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  S  C_  T
)
Distinct variable groups:    x, A    x, J    x, X
Allowed substitution hints:    S( x)    T( x)

Proof of Theorem concompclo
StepHypRef Expression
1 eqid 2283 . 2  |-  U. J  =  U. J
2 simp1 955 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  J  e.  (TopOn `  X ) )
3 inss1 3389 . . . . . . 7  |-  ( J  i^i  ( Clsd `  J
) )  C_  J
4 simp2 956 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  e.  ( J  i^i  ( Clsd `  J ) ) )
53, 4sseldi 3178 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  e.  J )
6 toponss 16667 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  J )  ->  T  C_  X )
72, 5, 6syl2anc 642 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  C_  X
)
8 simp3 957 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  A  e.  T )
97, 8sseldd 3181 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  A  e.  X )
10 concomp.2 . . . . 5  |-  S  = 
U. { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }
1110concompcld 17160 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  e.  ( Clsd `  J
) )
122, 9, 11syl2anc 642 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  S  e.  ( Clsd `  J )
)
131cldss 16766 . . 3  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  U. J
)
1412, 13syl 15 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  S  C_  U. J
)
1510concompcon 17158 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( Jt  S )  e.  Con )
162, 9, 15syl2anc 642 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  ( Jt  S
)  e.  Con )
1710concompid 17157 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  S )
182, 9, 17syl2anc 642 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  A  e.  S )
19 inelcm 3509 . . 3  |-  ( ( A  e.  T  /\  A  e.  S )  ->  ( T  i^i  S
)  =/=  (/) )
208, 18, 19syl2anc 642 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  ( T  i^i  S )  =/=  (/) )
21 inss2 3390 . . 3  |-  ( J  i^i  ( Clsd `  J
) )  C_  ( Clsd `  J )
2221, 4sseldi 3178 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  e.  ( Clsd `  J )
)
231, 14, 16, 5, 20, 22consubclo 17150 1  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  S  C_  T
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   ` cfv 5255  (class class class)co 5858   ↾t crest 13325  TopOnctopon 16632   Clsdccld 16753   Conccon 17137
This theorem is referenced by:  tgpconcompss  17796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-ntr 16757  df-cls 16758  df-con 17138
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