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Theorem restcldi 19857
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 996 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  B  e.  ( Clsd `  J
) )
2 dfss 3426 . . . . 5  |-  ( B 
C_  A  <->  B  =  ( B  i^i  A ) )
32biimpi 194 . . . 4  |-  ( B 
C_  A  ->  B  =  ( B  i^i  A ) )
433ad2ant3 1018 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  B  =  ( B  i^i  A ) )
5 ineq1 3631 . . . . 5  |-  ( v  =  B  ->  (
v  i^i  A )  =  ( B  i^i  A ) )
65eqeq2d 2414 . . . 4  |-  ( v  =  B  ->  ( B  =  ( v  i^i  A )  <->  B  =  ( B  i^i  A ) ) )
76rspcev 3157 . . 3  |-  ( ( B  e.  ( Clsd `  J )  /\  B  =  ( B  i^i  A ) )  ->  E. v  e.  ( Clsd `  J
) B  =  ( v  i^i  A ) )
81, 4, 7syl2anc 659 . 2  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  E. v  e.  ( Clsd `  J
) B  =  ( v  i^i  A ) )
9 cldrcl 19709 . . . 4  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
1093ad2ant2 1017 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  J  e.  Top )
11 simp1 995 . . 3  |-  ( ( A  C_  X  /\  B  e.  ( Clsd `  J )  /\  B  C_  A )  ->  A  C_  X )
12 restcldi.1 . . . 4  |-  X  = 
U. J
1312restcld 19856 . . 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 659 . 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 972    = wceq 1403    e. wcel 1840   E.wrex 2752    i^i cin 3410    C_ wss 3411   U.cuni 4188   ` cfv 5523  (class class class)co 6232   ↾t crest 14925   Topctop 19576   Clsdccld 19699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-oadd 7089  df-er 7266  df-en 7473  df-fin 7476  df-fi 7823  df-rest 14927  df-topgen 14948  df-top 19581  df-bases 19583  df-topon 19584  df-cld 19702
This theorem is referenced by:  txkgen  20335  qtoprest  20400  cnmpt2pc  21610  cnheiborlem  21636  abelth  23018  cvmliftlem10  29467
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