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Theorem concompclo 19802
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 2467 . 2  |-  U. J  =  U. J
2 simp1 996 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  J  e.  (TopOn `  X ) )
3 inss1 3723 . . . . . . 7  |-  ( J  i^i  ( Clsd `  J
) )  C_  J
4 simp2 997 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  e.  ( J  i^i  ( Clsd `  J ) ) )
53, 4sseldi 3507 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  e.  J )
6 toponss 19297 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  J )  ->  T  C_  X )
72, 5, 6syl2anc 661 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  C_  X
)
8 simp3 998 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  A  e.  T )
97, 8sseldd 3510 . . . 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 19801 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  e.  ( Clsd `  J
) )
122, 9, 11syl2anc 661 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  S  e.  ( Clsd `  J )
)
131cldss 19396 . . 3  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  U. J
)
1412, 13syl 16 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  S  C_  U. J
)
1510concompcon 19799 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( Jt  S )  e.  Con )
162, 9, 15syl2anc 661 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  ( Jt  S
)  e.  Con )
1710concompid 19798 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  S )
182, 9, 17syl2anc 661 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  A  e.  S )
19 inelcm 3886 . . 3  |-  ( ( A  e.  T  /\  A  e.  S )  ->  ( T  i^i  S
)  =/=  (/) )
208, 18, 19syl2anc 661 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  ( T  i^i  S )  =/=  (/) )
21 inss2 3724 . . 3  |-  ( J  i^i  ( Clsd `  J
) )  C_  ( Clsd `  J )
2221, 4sseldi 3507 . 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 19791 1  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  S  C_  T
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2821    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   U.cuni 4251   ` cfv 5594  (class class class)co 6295   ↾t crest 14692  TopOnctopon 19262   Clsdccld 19383   Conccon 19778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-oadd 7146  df-er 7323  df-en 7529  df-fin 7532  df-fi 7883  df-rest 14694  df-topgen 14715  df-top 19266  df-bases 19268  df-topon 19269  df-cld 19386  df-ntr 19387  df-cls 19388  df-con 19779
This theorem is referenced by:  tgpconcompss  20478
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