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Theorem consubclo 17367
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 2366 . . . 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 16980 . . . . . 6  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
53, 4syl 15 . . . . 5  |-  ( ph  ->  J  e.  Top )
6 consubclo.3 . . . . . 6  |-  ( ph  ->  A  C_  X )
7 consubclo.1 . . . . . . . 8  |-  X  = 
U. J
87topopn 16869 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  J )
95, 8syl 15 . . . . . 6  |-  ( ph  ->  X  e.  J )
10 ssexg 4262 . . . . . 6  |-  ( ( A  C_  X  /\  X  e.  J )  ->  A  e.  _V )
116, 9, 10syl2anc 642 . . . . 5  |-  ( ph  ->  A  e.  _V )
12 consubclo.5 . . . . 5  |-  ( ph  ->  B  e.  J )
13 elrestr 13543 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V  /\  B  e.  J )  ->  ( B  i^i  A )  e.  ( Jt  A ) )
145, 11, 12, 13syl3anc 1183 . . . 4  |-  ( ph  ->  ( B  i^i  A
)  e.  ( Jt  A ) )
15 consubclo.6 . . . 4  |-  ( ph  ->  ( B  i^i  A
)  =/=  (/) )
16 eqid 2366 . . . . . 6  |-  ( B  i^i  A )  =  ( B  i^i  A
)
17 ineq1 3451 . . . . . . . 8  |-  ( x  =  B  ->  (
x  i^i  A )  =  ( B  i^i  A ) )
1817eqeq2d 2377 . . . . . . 7  |-  ( x  =  B  ->  (
( B  i^i  A
)  =  ( x  i^i  A )  <->  ( B  i^i  A )  =  ( B  i^i  A ) ) )
1918rspcev 2969 . . . . . 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 ) )
203, 16, 19sylancl 643 . . . . 5  |-  ( ph  ->  E. x  e.  (
Clsd `  J )
( B  i^i  A
)  =  ( x  i^i  A ) )
217restcld 17120 . . . . . 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 ) ) )
225, 6, 21syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( B  i^i  A )  e.  ( Clsd `  ( Jt  A ) )  <->  E. x  e.  ( Clsd `  J
) ( B  i^i  A )  =  ( x  i^i  A ) ) )
2320, 22mpbird 223 . . . 4  |-  ( ph  ->  ( B  i^i  A
)  e.  ( Clsd `  ( Jt  A ) ) )
241, 2, 14, 15, 23conclo 17358 . . 3  |-  ( ph  ->  ( B  i^i  A
)  =  U. ( Jt  A ) )
257restuni 17110 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  =  U. ( Jt  A ) )
265, 6, 25syl2anc 642 . . 3  |-  ( ph  ->  A  =  U. ( Jt  A ) )
2724, 26eqtr4d 2401 . 2  |-  ( ph  ->  ( B  i^i  A
)  =  A )
28 dfss1 3461 . 2  |-  ( A 
C_  B  <->  ( B  i^i  A )  =  A )
2927, 28sylibr 203 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1647    e. wcel 1715    =/= wne 2529   E.wrex 2629   _Vcvv 2873    i^i cin 3237    C_ wss 3238   (/)c0 3543   U.cuni 3929   ` cfv 5358  (class class class)co 5981   ↾t crest 13535   Topctop 16848   Clsdccld 16970   Conccon 17354
This theorem is referenced by:  concn  17369  concompclo  17378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-recs 6530  df-rdg 6565  df-oadd 6625  df-er 6802  df-en 7007  df-fin 7010  df-fi 7312  df-rest 13537  df-topgen 13554  df-top 16853  df-bases 16855  df-topon 16856  df-cld 16973  df-con 17355
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