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Theorem concompcld 19180
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 18673 . . . . . 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 3548 . . . . . . . 8  |-  { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  C_  ~P X
5 sspwuni 4367 . . . . . . . 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 3497 . . . . . 6  |-  S  C_  X
8 toponuni 18674 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
98adantr 465 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  X  =  U. J )
107, 9syl5sseq 3515 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  S  C_ 
U. J )
11 eqid 2454 . . . . . 6  |-  U. J  =  U. J
1211clsss3 18805 . . . . 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 3504 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( cls `  J
) `  S )  C_  X )
1511sscls 18802 . . . . 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 19177 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  S )
1816, 17sseldd 3468 . . 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 19178 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( Jt  S )  e.  Con )
22 clscon 19176 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  ( Jt  S )  e.  Con )  ->  ( Jt  ( ( cls `  J ) `  S
) )  e.  Con )
2319, 20, 21, 22syl3anc 1219 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( Jt  ( ( cls `  J
) `  S )
)  e.  Con )
243concompss 19179 . . 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 1219 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( cls `  J
) `  S )  C_  S )
2611iscld4 18811 . . 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 1370    e. wcel 1758   {crab 2803    C_ wss 3439   ~Pcpw 3971   U.cuni 4202   ` cfv 5529  (class class class)co 6203   ↾t crest 14482   Topctop 18640  TopOnctopon 18641   Clsdccld 18762   clsccl 18764   Conccon 19157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-oadd 7037  df-er 7214  df-en 7424  df-fin 7427  df-fi 7776  df-rest 14484  df-topgen 14505  df-top 18645  df-bases 18647  df-topon 18648  df-cld 18765  df-ntr 18766  df-cls 18767  df-con 19158
This theorem is referenced by:  concompclo  19181
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