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Theorem restcldi 16960
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 956 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  B  e.  ( Clsd `  J
) )
2 dfss 3201 . . . . 5  |-  ( B 
C_  A  <->  B  =  ( B  i^i  A ) )
32biimpi 186 . . . 4  |-  ( B 
C_  A  ->  B  =  ( B  i^i  A ) )
433ad2ant3 978 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  B  =  ( B  i^i  A ) )
5 ineq1 3397 . . . . 5  |-  ( v  =  B  ->  (
v  i^i  A )  =  ( B  i^i  A ) )
65eqeq2d 2327 . . . 4  |-  ( v  =  B  ->  ( B  =  ( v  i^i  A )  <->  B  =  ( B  i^i  A ) ) )
76rspcev 2918 . . 3  |-  ( ( B  e.  ( Clsd `  J )  /\  B  =  ( B  i^i  A ) )  ->  E. v  e.  ( Clsd `  J
) B  =  ( v  i^i  A ) )
81, 4, 7syl2anc 642 . 2  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  E. v  e.  ( Clsd `  J
) B  =  ( v  i^i  A ) )
9 cldrcl 16819 . . . 4  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
1093ad2ant2 977 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  J  e.  Top )
11 simp1 955 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  A  C_  X )
12 restcldi.1 . . . 4  |-  X  = 
U. J
1312restcld 16959 . . 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 642 . 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 223 1  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  B  e.  ( Clsd `  ( Jt  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1633    e. wcel 1701   E.wrex 2578    i^i cin 3185    C_ wss 3186   U.cuni 3864   ` cfv 5292  (class class class)co 5900   ↾t crest 13374   Topctop 16687   Clsdccld 16809
This theorem is referenced by:  txkgen  17402  qtoprest  17464  cnmpt2pc  18479  cnheiborlem  18505  abelth  19870  cvmliftlem10  24109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-recs 6430  df-rdg 6465  df-oadd 6525  df-er 6702  df-en 6907  df-fin 6910  df-fi 7210  df-rest 13376  df-topgen 13393  df-top 16692  df-bases 16694  df-topon 16695  df-cld 16812
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