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Theorem cnrest 14212
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 2189 . . . . 5  |-  U. K  =  U. K
31, 2cnf 14181 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
43adantr 276 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  F : X --> U. K
)
5 simpr 110 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  C_  X )
64, 5fssresd 5411 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A ) : A --> U. K
)
7 cnvresima 5136 . . . 4  |-  ( `' ( F  |`  A )
" o )  =  ( ( `' F " o )  i^i  A
)
8 cntop1 14178 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
98adantr 276 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  J  e.  Top )
109adantr 276 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  J  e.  Top )
111topopn 13985 . . . . . . . 8  |-  ( J  e.  Top  ->  X  e.  J )
12 ssexg 4157 . . . . . . . . 9  |-  ( ( A  C_  X  /\  X  e.  J )  ->  A  e.  _V )
1312ancoms 268 . . . . . . . 8  |-  ( ( X  e.  J  /\  A  C_  X )  ->  A  e.  _V )
1411, 13sylan 283 . . . . . . 7  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  e.  _V )
158, 14sylan 283 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  e.  _V )
1615adantr 276 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  A  e.  _V )
17 cnima 14197 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  o  e.  K )  ->  ( `' F "
o )  e.  J
)
1817adantlr 477 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  ( `' F " o )  e.  J )
19 elrestr 12755 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V  /\  ( `' F " o )  e.  J )  -> 
( ( `' F " o )  i^i  A
)  e.  ( Jt  A ) )
2010, 16, 18, 19syl3anc 1249 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  (
( `' F "
o )  i^i  A
)  e.  ( Jt  A ) )
217, 20eqeltrid 2276 . . 3  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
2221ralrimiva 2563 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A. o  e.  K  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
231toptopon 13995 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
248, 23sylib 122 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  (TopOn `  X )
)
25 resttopon 14148 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )
2624, 25sylan 283 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( Jt  A )  e.  (TopOn `  A ) )
27 cntop2 14179 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
2827adantr 276 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  K  e.  Top )
292toptopon 13995 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
3028, 29sylib 122 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  K  e.  (TopOn `  U. K ) )
31 iscn 14174 . . 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 411 . 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 946 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 104    <-> wb 105    = wceq 1364    e. wcel 2160   A.wral 2468   _Vcvv 2752    i^i cin 3143    C_ wss 3144   U.cuni 3824   `'ccnv 4643    |` cres 4646   "cima 4647   -->wf 5231   ` cfv 5235  (class class class)co 5897   ↾t crest 12747   Topctop 13974  TopOnctopon 13987    Cn ccn 14162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-map 6677  df-rest 12749  df-topgen 12768  df-top 13975  df-topon 13988  df-bases 14020  df-cn 14165
This theorem is referenced by:  cnmpt1res  14273  cnmpt2res  14274  hmeores  14292
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