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Theorem cnrest 19659
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 2443 . . . . 5  |-  U. K  =  U. K
31, 2cnf 19620 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
43adantr 465 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  F : X --> U. K
)
5 simpr 461 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  C_  X )
64, 5fssresd 5742 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A ) : A --> U. K
)
7 cnvresima 5486 . . . 4  |-  ( `' ( F  |`  A )
" o )  =  ( ( `' F " o )  i^i  A
)
8 cntop1 19614 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
98adantr 465 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  J  e.  Top )
109adantr 465 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  J  e.  Top )
111topopn 19288 . . . . . . . 8  |-  ( J  e.  Top  ->  X  e.  J )
12 ssexg 4583 . . . . . . . . 9  |-  ( ( A  C_  X  /\  X  e.  J )  ->  A  e.  _V )
1312ancoms 453 . . . . . . . 8  |-  ( ( X  e.  J  /\  A  C_  X )  ->  A  e.  _V )
1411, 13sylan 471 . . . . . . 7  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  e.  _V )
158, 14sylan 471 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  e.  _V )
1615adantr 465 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  A  e.  _V )
17 cnima 19639 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  o  e.  K )  ->  ( `' F "
o )  e.  J
)
1817adantlr 714 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  ( `' F " o )  e.  J )
19 elrestr 14703 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V  /\  ( `' F " o )  e.  J )  -> 
( ( `' F " o )  i^i  A
)  e.  ( Jt  A ) )
2010, 16, 18, 19syl3anc 1229 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  (
( `' F "
o )  i^i  A
)  e.  ( Jt  A ) )
217, 20syl5eqel 2535 . . 3  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
2221ralrimiva 2857 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A. o  e.  K  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
231toptopon 19307 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
248, 23sylib 196 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  (TopOn `  X )
)
25 resttopon 19535 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )
2624, 25sylan 471 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( Jt  A )  e.  (TopOn `  A ) )
27 cntop2 19615 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
2827adantr 465 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  K  e.  Top )
292toptopon 19307 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
3028, 29sylib 196 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  K  e.  (TopOn `  U. K ) )
31 iscn 19609 . . 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 661 . 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 922 1  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095    i^i cin 3460    C_ wss 3461   U.cuni 4234   `'ccnv 4988    |` cres 4991   "cima 4992   -->wf 5574   ` cfv 5578  (class class class)co 6281   ↾t crest 14695   Topctop 19267  TopOnctopon 19268    Cn ccn 19598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-fin 7522  df-fi 7873  df-rest 14697  df-topgen 14718  df-top 19272  df-bases 19274  df-topon 19275  df-cn 19601
This theorem is referenced by:  resthauslem  19737  imacmp  19770  conima  19799  kgencn2  19931  kgencn3  19932  xkopjcn  20030  cnmpt1res  20050  cnmpt2res  20051  qtoprest  20091  hmeores  20145  ftalem3  23220  rmulccn  27783  raddcn  27784  xrge0mulc1cn  27796  rrhre  27872  cvmliftmolem1  28599  cvmlift2lem9a  28621  cvmlift2lem9  28629  areacirclem2  30083  ivthALT  30128  cnres2  30234  stoweidlem28  31699  dirkercncflem2  31775
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