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Theorem concompss 17159
Description: The connected component containing  A is a superset of any other connected set containing  A. (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
concompss  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  C_  S
)
Distinct variable groups:    x, A    x, J    x, X
Allowed substitution hints:    S( x)    T( x)

Proof of Theorem concompss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simp1 955 . . . . 5  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  C_  X
)
2 contop 17143 . . . . . . 7  |-  ( ( Jt  T )  e.  Con  ->  ( Jt  T )  e.  Top )
323ad2ant3 978 . . . . . 6  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  ( Jt  T )  e.  Top )
4 restrcl 16888 . . . . . . 7  |-  ( ( Jt  T )  e.  Top  ->  ( J  e.  _V  /\  T  e.  _V )
)
54simprd 449 . . . . . 6  |-  ( ( Jt  T )  e.  Top  ->  T  e.  _V )
6 elpwg 3632 . . . . . 6  |-  ( T  e.  _V  ->  ( T  e.  ~P X  <->  T 
C_  X ) )
73, 5, 63syl 18 . . . . 5  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  ( T  e. 
~P X  <->  T  C_  X
) )
81, 7mpbird 223 . . . 4  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  e.  ~P X )
9 3simpc 954 . . . 4  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  ( A  e.  T  /\  ( Jt  T )  e.  Con )
)
10 eleq2 2344 . . . . . 6  |-  ( y  =  T  ->  ( A  e.  y  <->  A  e.  T ) )
11 oveq2 5866 . . . . . . 7  |-  ( y  =  T  ->  ( Jt  y )  =  ( Jt  T ) )
1211eleq1d 2349 . . . . . 6  |-  ( y  =  T  ->  (
( Jt  y )  e. 
Con 
<->  ( Jt  T )  e.  Con ) )
1310, 12anbi12d 691 . . . . 5  |-  ( y  =  T  ->  (
( A  e.  y  /\  ( Jt  y )  e.  Con )  <->  ( A  e.  T  /\  ( Jt  T )  e.  Con ) ) )
14 eleq2 2344 . . . . . . 7  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
15 oveq2 5866 . . . . . . . 8  |-  ( x  =  y  ->  ( Jt  x )  =  ( Jt  y ) )
1615eleq1d 2349 . . . . . . 7  |-  ( x  =  y  ->  (
( Jt  x )  e.  Con  <->  ( Jt  y )  e.  Con ) )
1714, 16anbi12d 691 . . . . . 6  |-  ( x  =  y  ->  (
( A  e.  x  /\  ( Jt  x )  e.  Con ) 
<->  ( A  e.  y  /\  ( Jt  y )  e.  Con ) ) )
1817cbvrabv 2787 . . . . 5  |-  { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  =  {
y  e.  ~P X  |  ( A  e.  y  /\  ( Jt  y )  e.  Con ) }
1913, 18elrab2 2925 . . . 4  |-  ( T  e.  { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  <->  ( T  e. 
~P X  /\  ( A  e.  T  /\  ( Jt  T )  e.  Con ) ) )
208, 9, 19sylanbrc 645 . . 3  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  e.  {
x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) } )
21 elssuni 3855 . . 3  |-  ( T  e.  { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }  ->  T  C_  U. { x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) } )
2220, 21syl 15 . 2  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  C_  U. {
x  e.  ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) } )
23 concomp.2 . 2  |-  S  = 
U. { x  e. 
~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) }
2422, 23syl6sseqr 3225 1  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  C_  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   U.cuni 3827  (class class class)co 5858   ↾t crest 13325   Topctop 16631   Conccon 17137
This theorem is referenced by:  concompcld  17160  tgpconcompeqg  17794  tgpconcomp  17795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-rest 13327  df-top 16636  df-con 17138
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