MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  restcldi Structured version   Unicode version

Theorem restcldi 17237
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 958 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  B  e.  ( Clsd `  J
) )
2 dfss 3335 . . . . 5  |-  ( B 
C_  A  <->  B  =  ( B  i^i  A ) )
32biimpi 187 . . . 4  |-  ( B 
C_  A  ->  B  =  ( B  i^i  A ) )
433ad2ant3 980 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  B  =  ( B  i^i  A ) )
5 ineq1 3535 . . . . 5  |-  ( v  =  B  ->  (
v  i^i  A )  =  ( B  i^i  A ) )
65eqeq2d 2447 . . . 4  |-  ( v  =  B  ->  ( B  =  ( v  i^i  A )  <->  B  =  ( B  i^i  A ) ) )
76rspcev 3052 . . 3  |-  ( ( B  e.  ( Clsd `  J )  /\  B  =  ( B  i^i  A ) )  ->  E. v  e.  ( Clsd `  J
) B  =  ( v  i^i  A ) )
81, 4, 7syl2anc 643 . 2  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  E. v  e.  ( Clsd `  J
) B  =  ( v  i^i  A ) )
9 cldrcl 17090 . . . 4  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
1093ad2ant2 979 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  J  e.  Top )
11 simp1 957 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  A  C_  X )
12 restcldi.1 . . . 4  |-  X  = 
U. J
1312restcld 17236 . . 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 643 . 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 224 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 177    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706    i^i cin 3319    C_ wss 3320   U.cuni 4015   ` cfv 5454  (class class class)co 6081   ↾t crest 13648   Topctop 16958   Clsdccld 17080
This theorem is referenced by:  txkgen  17684  qtoprest  17749  cnmpt2pc  18953  cnheiborlem  18979  abelth  20357  cvmliftlem10  24981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-oadd 6728  df-er 6905  df-en 7110  df-fin 7113  df-fi 7416  df-rest 13650  df-topgen 13667  df-top 16963  df-bases 16965  df-topon 16966  df-cld 17083
  Copyright terms: Public domain W3C validator