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Theorem consubclo 19043
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 2443 . . . 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 18645 . . . . . 6  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
53, 4syl 16 . . . . 5  |-  ( ph  ->  J  e.  Top )
6 consubclo.1 . . . . . . . 8  |-  X  = 
U. J
76topopn 18534 . . . . . . 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 4454 . . . . 5  |-  ( ph  ->  A  e.  _V )
11 consubclo.5 . . . . 5  |-  ( ph  ->  B  e.  J )
12 elrestr 14382 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V  /\  B  e.  J )  ->  ( B  i^i  A )  e.  ( Jt  A ) )
135, 10, 11, 12syl3anc 1218 . . . 4  |-  ( ph  ->  ( B  i^i  A
)  e.  ( Jt  A ) )
14 consubclo.6 . . . 4  |-  ( ph  ->  ( B  i^i  A
)  =/=  (/) )
15 eqid 2443 . . . . . 6  |-  ( B  i^i  A )  =  ( B  i^i  A
)
16 ineq1 3560 . . . . . . . 8  |-  ( x  =  B  ->  (
x  i^i  A )  =  ( B  i^i  A ) )
1716eqeq2d 2454 . . . . . . 7  |-  ( x  =  B  ->  (
( B  i^i  A
)  =  ( x  i^i  A )  <->  ( B  i^i  A )  =  ( B  i^i  A ) ) )
1817rspcev 3088 . . . . . 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 18791 . . . . . 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 19034 . . 3  |-  ( ph  ->  ( B  i^i  A
)  =  U. ( Jt  A ) )
246restuni 18781 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  =  U. ( Jt  A ) )
255, 9, 24syl2anc 661 . . 3  |-  ( ph  ->  A  =  U. ( Jt  A ) )
2623, 25eqtr4d 2478 . 2  |-  ( ph  ->  ( B  i^i  A
)  =  A )
27 dfss1 3570 . 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 1369    e. wcel 1756    =/= wne 2620   E.wrex 2731   _Vcvv 2987    i^i cin 3342    C_ wss 3343   (/)c0 3652   U.cuni 4106   ` cfv 5433  (class class class)co 6106   ↾t crest 14374   Topctop 18513   Clsdccld 18635   Conccon 19030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-oadd 6939  df-er 7116  df-en 7326  df-fin 7329  df-fi 7676  df-rest 14376  df-topgen 14397  df-top 18518  df-bases 18520  df-topon 18521  df-cld 18638  df-con 19031
This theorem is referenced by:  concn  19045  concompclo  19054
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