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Theorem restcldi 18752
Description: A closed set is closed in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
restcldi.1  |-  X  = 
U. J
Assertion
Ref Expression
restcldi  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  B  e.  ( Clsd `  ( Jt  A ) ) )

Proof of Theorem restcldi
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 simp2 989 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  B  e.  ( Clsd `  J
) )
2 dfss 3338 . . . . 5  |-  ( B 
C_  A  <->  B  =  ( B  i^i  A ) )
32biimpi 194 . . . 4  |-  ( B 
C_  A  ->  B  =  ( B  i^i  A ) )
433ad2ant3 1011 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  B  =  ( B  i^i  A ) )
5 ineq1 3540 . . . . 5  |-  ( v  =  B  ->  (
v  i^i  A )  =  ( B  i^i  A ) )
65eqeq2d 2449 . . . 4  |-  ( v  =  B  ->  ( B  =  ( v  i^i  A )  <->  B  =  ( B  i^i  A ) ) )
76rspcev 3068 . . 3  |-  ( ( B  e.  ( Clsd `  J )  /\  B  =  ( B  i^i  A ) )  ->  E. v  e.  ( Clsd `  J
) B  =  ( v  i^i  A ) )
81, 4, 7syl2anc 661 . 2  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  E. v  e.  ( Clsd `  J
) B  =  ( v  i^i  A ) )
9 cldrcl 18605 . . . 4  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
1093ad2ant2 1010 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  J  e.  Top )
11 simp1 988 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  A  C_  X )
12 restcldi.1 . . . 4  |-  X  = 
U. J
1312restcld 18751 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( B  e.  (
Clsd `  ( Jt  A
) )  <->  E. v  e.  ( Clsd `  J
) B  =  ( v  i^i  A ) ) )
1410, 11, 13syl2anc 661 . 2  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  ( B  e.  ( Clsd `  ( Jt  A ) )  <->  E. v  e.  ( Clsd `  J
) B  =  ( v  i^i  A ) ) )
158, 14mpbird 232 1  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  B  e.  ( Clsd `  ( Jt  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2711    i^i cin 3322    C_ wss 3323   U.cuni 4086   ` cfv 5413  (class class class)co 6086   ↾t crest 14351   Topctop 18473   Clsdccld 18595
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-oadd 6916  df-er 7093  df-en 7303  df-fin 7306  df-fi 7653  df-rest 14353  df-topgen 14374  df-top 18478  df-bases 18480  df-topon 18481  df-cld 18598
This theorem is referenced by:  txkgen  19200  qtoprest  19265  cnmpt2pc  20475  cnheiborlem  20501  abelth  21881  cvmliftlem10  27135
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