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Theorem consubclo 17492
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 2438 . . . 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 17095 . . . . . 6  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
53, 4syl 16 . . . . 5  |-  ( ph  ->  J  e.  Top )
6 consubclo.1 . . . . . . . 8  |-  X  = 
U. J
76topopn 16984 . . . . . . 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 4353 . . . . 5  |-  ( ph  ->  A  e.  _V )
11 consubclo.5 . . . . 5  |-  ( ph  ->  B  e.  J )
12 elrestr 13661 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V  /\  B  e.  J )  ->  ( B  i^i  A )  e.  ( Jt  A ) )
135, 10, 11, 12syl3anc 1185 . . . 4  |-  ( ph  ->  ( B  i^i  A
)  e.  ( Jt  A ) )
14 consubclo.6 . . . 4  |-  ( ph  ->  ( B  i^i  A
)  =/=  (/) )
15 eqid 2438 . . . . . 6  |-  ( B  i^i  A )  =  ( B  i^i  A
)
16 ineq1 3537 . . . . . . . 8  |-  ( x  =  B  ->  (
x  i^i  A )  =  ( B  i^i  A ) )
1716eqeq2d 2449 . . . . . . 7  |-  ( x  =  B  ->  (
( B  i^i  A
)  =  ( x  i^i  A )  <->  ( B  i^i  A )  =  ( B  i^i  A ) ) )
1817rspcev 3054 . . . . . 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 645 . . . . 5  |-  ( ph  ->  E. x  e.  (
Clsd `  J )
( B  i^i  A
)  =  ( x  i^i  A ) )
206restcld 17241 . . . . . 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 644 . . . . 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 225 . . . 4  |-  ( ph  ->  ( B  i^i  A
)  e.  ( Clsd `  ( Jt  A ) ) )
231, 2, 13, 14, 22conclo 17483 . . 3  |-  ( ph  ->  ( B  i^i  A
)  =  U. ( Jt  A ) )
246restuni 17231 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  =  U. ( Jt  A ) )
255, 9, 24syl2anc 644 . . 3  |-  ( ph  ->  A  =  U. ( Jt  A ) )
2623, 25eqtr4d 2473 . 2  |-  ( ph  ->  ( B  i^i  A
)  =  A )
27 dfss1 3547 . 2  |-  ( A 
C_  B  <->  ( B  i^i  A )  =  A )
2826, 27sylibr 205 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   _Vcvv 2958    i^i cin 3321    C_ wss 3322   (/)c0 3630   U.cuni 4017   ` cfv 5457  (class class class)co 6084   ↾t crest 13653   Topctop 16963   Clsdccld 17085   Conccon 17479
This theorem is referenced by:  concn  17494  concompclo  17503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-oadd 6731  df-er 6908  df-en 7113  df-fin 7116  df-fi 7419  df-rest 13655  df-topgen 13672  df-top 16968  df-bases 16970  df-topon 16971  df-cld 17088  df-con 17480
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