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Theorem consubclo 19793
Description: If a clopen set meets a connected subspace, it must contain the entire subspace. (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypotheses
Ref Expression
consubclo.1  |-  X  = 
U. J
consubclo.3  |-  ( ph  ->  A  C_  X )
consubclo.4  |-  ( ph  ->  ( Jt  A )  e.  Con )
consubclo.5  |-  ( ph  ->  B  e.  J )
consubclo.6  |-  ( ph  ->  ( B  i^i  A
)  =/=  (/) )
consubclo.7  |-  ( ph  ->  B  e.  ( Clsd `  J ) )
Assertion
Ref Expression
consubclo  |-  ( ph  ->  A  C_  B )

Proof of Theorem consubclo
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . 4  |-  U. ( Jt  A )  =  U. ( Jt  A )
2 consubclo.4 . . . 4  |-  ( ph  ->  ( Jt  A )  e.  Con )
3 consubclo.7 . . . . . 6  |-  ( ph  ->  B  e.  ( Clsd `  J ) )
4 cldrcl 19395 . . . . . 6  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
53, 4syl 16 . . . . 5  |-  ( ph  ->  J  e.  Top )
6 consubclo.1 . . . . . . . 8  |-  X  = 
U. J
76topopn 19284 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  J )
85, 7syl 16 . . . . . 6  |-  ( ph  ->  X  e.  J )
9 consubclo.3 . . . . . 6  |-  ( ph  ->  A  C_  X )
108, 9ssexd 4600 . . . . 5  |-  ( ph  ->  A  e.  _V )
11 consubclo.5 . . . . 5  |-  ( ph  ->  B  e.  J )
12 elrestr 14701 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V  /\  B  e.  J )  ->  ( B  i^i  A )  e.  ( Jt  A ) )
135, 10, 11, 12syl3anc 1228 . . . 4  |-  ( ph  ->  ( B  i^i  A
)  e.  ( Jt  A ) )
14 consubclo.6 . . . 4  |-  ( ph  ->  ( B  i^i  A
)  =/=  (/) )
15 eqid 2467 . . . . . 6  |-  ( B  i^i  A )  =  ( B  i^i  A
)
16 ineq1 3698 . . . . . . . 8  |-  ( x  =  B  ->  (
x  i^i  A )  =  ( B  i^i  A ) )
1716eqeq2d 2481 . . . . . . 7  |-  ( x  =  B  ->  (
( B  i^i  A
)  =  ( x  i^i  A )  <->  ( B  i^i  A )  =  ( B  i^i  A ) ) )
1817rspcev 3219 . . . . . 6  |-  ( ( B  e.  ( Clsd `  J )  /\  ( B  i^i  A )  =  ( B  i^i  A
) )  ->  E. x  e.  ( Clsd `  J
) ( B  i^i  A )  =  ( x  i^i  A ) )
193, 15, 18sylancl 662 . . . . 5  |-  ( ph  ->  E. x  e.  (
Clsd `  J )
( B  i^i  A
)  =  ( x  i^i  A ) )
206restcld 19541 . . . . . 6  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( B  i^i  A )  e.  ( Clsd `  ( Jt  A ) )  <->  E. x  e.  ( Clsd `  J
) ( B  i^i  A )  =  ( x  i^i  A ) ) )
215, 9, 20syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( B  i^i  A )  e.  ( Clsd `  ( Jt  A ) )  <->  E. x  e.  ( Clsd `  J
) ( B  i^i  A )  =  ( x  i^i  A ) ) )
2219, 21mpbird 232 . . . 4  |-  ( ph  ->  ( B  i^i  A
)  e.  ( Clsd `  ( Jt  A ) ) )
231, 2, 13, 14, 22conclo 19784 . . 3  |-  ( ph  ->  ( B  i^i  A
)  =  U. ( Jt  A ) )
246restuni 19531 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  =  U. ( Jt  A ) )
255, 9, 24syl2anc 661 . . 3  |-  ( ph  ->  A  =  U. ( Jt  A ) )
2623, 25eqtr4d 2511 . 2  |-  ( ph  ->  ( B  i^i  A
)  =  A )
27 dfss1 3708 . 2  |-  ( A 
C_  B  <->  ( B  i^i  A )  =  A )
2826, 27sylibr 212 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   _Vcvv 3118    i^i cin 3480    C_ wss 3481   (/)c0 3790   U.cuni 4251   ` cfv 5594  (class class class)co 6295   ↾t crest 14693   Topctop 19263   Clsdccld 19385   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-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-con 19781
This theorem is referenced by:  concn  19795  concompclo  19804
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