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Theorem cnrest 12185
Description: Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.)
Hypothesis
Ref Expression
cnrest.1  |-  X  = 
U. J
Assertion
Ref Expression
cnrest  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )

Proof of Theorem cnrest
Dummy variable  o is distinct from all other variables.
StepHypRef Expression
1 cnrest.1 . . . . 5  |-  X  = 
U. J
2 eqid 2100 . . . . 5  |-  U. K  =  U. K
31, 2cnf 12154 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
43adantr 272 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  F : X --> U. K
)
5 simpr 109 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  C_  X )
64, 5fssresd 5235 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A ) : A --> U. K
)
7 cnvresima 4964 . . . 4  |-  ( `' ( F  |`  A )
" o )  =  ( ( `' F " o )  i^i  A
)
8 cntop1 12151 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
98adantr 272 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  J  e.  Top )
109adantr 272 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  J  e.  Top )
111topopn 11957 . . . . . . . 8  |-  ( J  e.  Top  ->  X  e.  J )
12 ssexg 4007 . . . . . . . . 9  |-  ( ( A  C_  X  /\  X  e.  J )  ->  A  e.  _V )
1312ancoms 266 . . . . . . . 8  |-  ( ( X  e.  J  /\  A  C_  X )  ->  A  e.  _V )
1411, 13sylan 279 . . . . . . 7  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  e.  _V )
158, 14sylan 279 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  e.  _V )
1615adantr 272 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  A  e.  _V )
17 cnima 12170 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  o  e.  K )  ->  ( `' F "
o )  e.  J
)
1817adantlr 464 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  ( `' F " o )  e.  J )
19 elrestr 11910 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V  /\  ( `' F " o )  e.  J )  -> 
( ( `' F " o )  i^i  A
)  e.  ( Jt  A ) )
2010, 16, 18, 19syl3anc 1184 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  (
( `' F "
o )  i^i  A
)  e.  ( Jt  A ) )
217, 20syl5eqel 2186 . . 3  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
2221ralrimiva 2464 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A. o  e.  K  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
231toptopon 11967 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
248, 23sylib 121 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  (TopOn `  X )
)
25 resttopon 12122 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )
2624, 25sylan 279 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( Jt  A )  e.  (TopOn `  A ) )
27 cntop2 12152 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
2827adantr 272 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  K  e.  Top )
292toptopon 11967 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
3028, 29sylib 121 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  K  e.  (TopOn `  U. K ) )
31 iscn 12147 . . 3  |-  ( ( ( Jt  A )  e.  (TopOn `  A )  /\  K  e.  (TopOn `  U. K ) )  ->  ( ( F  |`  A )  e.  ( ( Jt  A )  Cn  K )  <->  ( ( F  |`  A ) : A --> U. K  /\  A. o  e.  K  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) ) ) )
3226, 30, 31syl2anc 406 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( ( F  |`  A )  e.  ( ( Jt  A )  Cn  K
)  <->  ( ( F  |`  A ) : A --> U. K  /\  A. o  e.  K  ( `' ( F  |`  A )
" o )  e.  ( Jt  A ) ) ) )
336, 22, 32mpbir2and 896 1  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1299    e. wcel 1448   A.wral 2375   _Vcvv 2641    i^i cin 3020    C_ wss 3021   U.cuni 3683   `'ccnv 4476    |` cres 4479   "cima 4480   -->wf 5055   ` cfv 5059  (class class class)co 5706   ↾t crest 11902   Topctop 11946  TopOnctopon 11959    Cn ccn 12136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-map 6474  df-rest 11904  df-topgen 11923  df-top 11947  df-topon 11960  df-bases 11992  df-cn 12139
This theorem is referenced by:  cnmpt1res  12246  cnmpt2res  12247
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