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Theorem concompclo 19061
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 2443 . 2  |-  U. J  =  U. J
2 simp1 988 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  J  e.  (TopOn `  X ) )
3 inss1 3591 . . . . . . 7  |-  ( J  i^i  ( Clsd `  J
) )  C_  J
4 simp2 989 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  e.  ( J  i^i  ( Clsd `  J ) ) )
53, 4sseldi 3375 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  T  e.  J )
6 toponss 18556 . . . . . 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 990 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  T  e.  ( J  i^i  ( Clsd `  J ) )  /\  A  e.  T
)  ->  A  e.  T )
97, 8sseldd 3378 . . . 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 19060 . . . 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 18655 . . 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 19058 . . 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 19057 . . . 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 3754 . . 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 3592 . . 3  |-  ( J  i^i  ( Clsd `  J
) )  C_  ( Clsd `  J )
2221, 4sseldi 3375 . 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 19050 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620   {crab 2740    i^i cin 3348    C_ wss 3349   (/)c0 3658   ~Pcpw 3881   U.cuni 4112   ` cfv 5439  (class class class)co 6112   ↾t crest 14380  TopOnctopon 18521   Clsdccld 18642   Conccon 19037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-oadd 6945  df-er 7122  df-en 7332  df-fin 7335  df-fi 7682  df-rest 14382  df-topgen 14403  df-top 18525  df-bases 18527  df-topon 18528  df-cld 18645  df-ntr 18646  df-cls 18647  df-con 19038
This theorem is referenced by:  tgpconcompss  19706
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