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Theorem concompss 17497
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 958 . . . . 5  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  C_  X
)
2 contop 17481 . . . . . . 7  |-  ( ( Jt  T )  e.  Con  ->  ( Jt  T )  e.  Top )
323ad2ant3 981 . . . . . 6  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  ( Jt  T )  e.  Top )
4 restrcl 17222 . . . . . . 7  |-  ( ( Jt  T )  e.  Top  ->  ( J  e.  _V  /\  T  e.  _V )
)
54simprd 451 . . . . . 6  |-  ( ( Jt  T )  e.  Top  ->  T  e.  _V )
6 elpwg 3807 . . . . . 6  |-  ( T  e.  _V  ->  ( T  e.  ~P X  <->  T 
C_  X ) )
73, 5, 63syl 19 . . . . 5  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  ( T  e. 
~P X  <->  T  C_  X
) )
81, 7mpbird 225 . . . 4  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  T  e.  ~P X )
9 3simpc 957 . . . 4  |-  ( ( T  C_  X  /\  A  e.  T  /\  ( Jt  T )  e.  Con )  ->  ( A  e.  T  /\  ( Jt  T )  e.  Con )
)
10 eleq2 2498 . . . . . 6  |-  ( y  =  T  ->  ( A  e.  y  <->  A  e.  T ) )
11 oveq2 6090 . . . . . . 7  |-  ( y  =  T  ->  ( Jt  y )  =  ( Jt  T ) )
1211eleq1d 2503 . . . . . 6  |-  ( y  =  T  ->  (
( Jt  y )  e. 
Con 
<->  ( Jt  T )  e.  Con ) )
1310, 12anbi12d 693 . . . . 5  |-  ( y  =  T  ->  (
( A  e.  y  /\  ( Jt  y )  e.  Con )  <->  ( A  e.  T  /\  ( Jt  T )  e.  Con ) ) )
14 eleq2 2498 . . . . . . 7  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
15 oveq2 6090 . . . . . . . 8  |-  ( x  =  y  ->  ( Jt  x )  =  ( Jt  y ) )
1615eleq1d 2503 . . . . . . 7  |-  ( x  =  y  ->  (
( Jt  x )  e.  Con  <->  ( Jt  y )  e.  Con ) )
1714, 16anbi12d 693 . . . . . 6  |-  ( x  =  y  ->  (
( A  e.  x  /\  ( Jt  x )  e.  Con ) 
<->  ( A  e.  y  /\  ( Jt  y )  e.  Con ) ) )
1817cbvrabv 2956 . . . . 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 3095 . . . 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 647 . . 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 4044 . . 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 16 . 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 3396 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {crab 2710   _Vcvv 2957    C_ wss 3321   ~Pcpw 3800   U.cuni 4016  (class class class)co 6082   ↾t crest 13649   Topctop 16959   Conccon 17475
This theorem is referenced by:  concompcld  17498  tgpconcompeqg  18142  tgpconcomp  18143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-rest 13651  df-top 16964  df-con 17476
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