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Theorem concompcld 19803
Description: The connected component containing  A is a closed set. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypothesis
Ref Expression
concomp.2  |-  S  = 
U. { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }
Assertion
Ref Expression
concompcld  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  e.  ( Clsd `  J
) )
Distinct variable groups:    x, A    x, J    x, X
Allowed substitution hint:    S( x)

Proof of Theorem concompcld
StepHypRef Expression
1 topontop 19296 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
21adantr 465 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  J  e.  Top )
3 concomp.2 . . . . . . 7  |-  S  = 
U. { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }
4 ssrab2 3590 . . . . . . . 8  |-  { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  C_  ~P X
5 sspwuni 4417 . . . . . . . 8  |-  ( { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  C_  ~P X  <->  U. { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  C_  X )
64, 5mpbi 208 . . . . . . 7  |-  U. {
x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  C_  X
73, 6eqsstri 3539 . . . . . 6  |-  S  C_  X
8 toponuni 19297 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
98adantr 465 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  X  =  U. J )
107, 9syl5sseq 3557 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  C_ 
U. J )
11 eqid 2467 . . . . . 6  |-  U. J  =  U. J
1211clsss3 19428 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  ( ( cls `  J ) `  S
)  C_  U. J )
132, 10, 12syl2anc 661 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( cls `  J
) `  S )  C_ 
U. J )
1413, 9sseqtr4d 3546 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( cls `  J
) `  S )  C_  X )
1511sscls 19425 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  S  C_  (
( cls `  J
) `  S )
)
162, 10, 15syl2anc 661 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  C_  ( ( cls `  J
) `  S )
)
173concompid 19800 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  S )
1816, 17sseldd 3510 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  ( ( cls `  J
) `  S )
)
19 simpl 457 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
207a1i 11 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  C_  X )
213concompcon 19801 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( Jt  S )  e.  Con )
22 clscon 19799 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  ( Jt  S )  e.  Con )  ->  ( Jt  ( ( cls `  J ) `  S
) )  e.  Con )
2319, 20, 21, 22syl3anc 1228 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( Jt  ( ( cls `  J
) `  S )
)  e.  Con )
243concompss 19802 . . 3  |-  ( ( ( ( cls `  J
) `  S )  C_  X  /\  A  e.  ( ( cls `  J
) `  S )  /\  ( Jt  ( ( cls `  J ) `  S
) )  e.  Con )  ->  ( ( cls `  J ) `  S
)  C_  S )
2514, 18, 23, 24syl3anc 1228 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( cls `  J
) `  S )  C_  S )
2611iscld4 19434 . . 3  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  ( S  e.  ( Clsd `  J
)  <->  ( ( cls `  J ) `  S
)  C_  S )
)
272, 10, 26syl2anc 661 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( S  e.  ( Clsd `  J )  <->  ( ( cls `  J ) `  S )  C_  S
) )
2825, 27mpbird 232 1  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  e.  ( Clsd `  J
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2821    C_ wss 3481   ~Pcpw 4016   U.cuni 4251   ` cfv 5594  (class class class)co 6295   ↾t crest 14693   Topctop 19263  TopOnctopon 19264   Clsdccld 19385   clsccl 19387   Conccon 19780
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 14695  df-topgen 14716  df-top 19268  df-bases 19270  df-topon 19271  df-cld 19388  df-ntr 19389  df-cls 19390  df-con 19781
This theorem is referenced by:  concompclo  19804
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