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Theorem consubclo 17448
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 2412 . . . 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 17053 . . . . . 6  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
53, 4syl 16 . . . . 5  |-  ( ph  ->  J  e.  Top )
6 consubclo.1 . . . . . . . 8  |-  X  = 
U. J
76topopn 16942 . . . . . . 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 4318 . . . . 5  |-  ( ph  ->  A  e.  _V )
11 consubclo.5 . . . . 5  |-  ( ph  ->  B  e.  J )
12 elrestr 13619 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V  /\  B  e.  J )  ->  ( B  i^i  A )  e.  ( Jt  A ) )
135, 10, 11, 12syl3anc 1184 . . . 4  |-  ( ph  ->  ( B  i^i  A
)  e.  ( Jt  A ) )
14 consubclo.6 . . . 4  |-  ( ph  ->  ( B  i^i  A
)  =/=  (/) )
15 eqid 2412 . . . . . 6  |-  ( B  i^i  A )  =  ( B  i^i  A
)
16 ineq1 3503 . . . . . . . 8  |-  ( x  =  B  ->  (
x  i^i  A )  =  ( B  i^i  A ) )
1716eqeq2d 2423 . . . . . . 7  |-  ( x  =  B  ->  (
( B  i^i  A
)  =  ( x  i^i  A )  <->  ( B  i^i  A )  =  ( B  i^i  A ) ) )
1817rspcev 3020 . . . . . 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 644 . . . . 5  |-  ( ph  ->  E. x  e.  (
Clsd `  J )
( B  i^i  A
)  =  ( x  i^i  A ) )
206restcld 17198 . . . . . 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 643 . . . . 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 224 . . . 4  |-  ( ph  ->  ( B  i^i  A
)  e.  ( Clsd `  ( Jt  A ) ) )
231, 2, 13, 14, 22conclo 17439 . . 3  |-  ( ph  ->  ( B  i^i  A
)  =  U. ( Jt  A ) )
246restuni 17188 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  =  U. ( Jt  A ) )
255, 9, 24syl2anc 643 . . 3  |-  ( ph  ->  A  =  U. ( Jt  A ) )
2623, 25eqtr4d 2447 . 2  |-  ( ph  ->  ( B  i^i  A
)  =  A )
27 dfss1 3513 . 2  |-  ( A 
C_  B  <->  ( B  i^i  A )  =  A )
2826, 27sylibr 204 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675   _Vcvv 2924    i^i cin 3287    C_ wss 3288   (/)c0 3596   U.cuni 3983   ` cfv 5421  (class class class)co 6048   ↾t crest 13611   Topctop 16921   Clsdccld 17043   Conccon 17435
This theorem is referenced by:  concn  17450  concompclo  17459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-oadd 6695  df-er 6872  df-en 7077  df-fin 7080  df-fi 7382  df-rest 13613  df-topgen 13630  df-top 16926  df-bases 16928  df-topon 16929  df-cld 17046  df-con 17436
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