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Theorem conncompclo 21238
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  = 
U. { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e. Conn ) }
Assertion
Ref Expression
conncompclo  |-  ( ( 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 conncompclo
StepHypRef Expression
1 eqid 2622 . 2  |-  U. J  =  U. J
2 simp1 1061 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  J  e.  (TopOn `  X ) )
3 inss1 3833 . . . . . . 7  |-  ( J  i^i  ( Clsd `  J
) )  C_  J
4 simp2 1062 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  e.  ( J  i^i  ( Clsd `  J ) ) )
53, 4sseldi 3601 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  e.  J )
6 toponss 20731 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  J )  ->  T  C_  X )
72, 5, 6syl2anc 693 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  C_  X
)
8 simp3 1063 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  A  e.  T )
97, 8sseldd 3604 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  A  e.  X )
10 conncomp.2 . . . . 5  |-  S  = 
U. { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e. Conn ) }
1110conncompcld 21237 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  e.  ( Clsd `  J
) )
122, 9, 11syl2anc 693 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  S  e.  ( Clsd `  J )
)
131cldss 20833 . . 3  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  U. J
)
1412, 13syl 17 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  S  C_  U. J
)
1510conncompconn 21235 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( Jt  S )  e. Conn )
162, 9, 15syl2anc 693 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  ( Jt  S
)  e. Conn )
1710conncompid 21234 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  S )
182, 9, 17syl2anc 693 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  A  e.  S )
19 inelcm 4032 . . 3  |-  ( ( A  e.  T  /\  A  e.  S )  ->  ( T  i^i  S
)  =/=  (/) )
208, 18, 19syl2anc 693 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  ( T  i^i  S )  =/=  (/) )
21 inss2 3834 . . 3  |-  ( J  i^i  ( Clsd `  J
) )  C_  ( Clsd `  J )
2221, 4sseldi 3601 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  e.  ( Clsd `  J )
)
231, 14, 16, 5, 20, 22connsubclo 21227 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 384    /\ w3a 1037    = wceq 1483    e. wcel 1990    =/= wne 2794   {crab 2916    i^i cin 3573    C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   ` cfv 5888  (class class class)co 6650   ↾t crest 16081  TopOnctopon 20715   Clsdccld 20820  Conncconn 21214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-conn 21215
This theorem is referenced by:  tgpconncompss  21917
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