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Theorem cnrest 17341
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 . . . . . . 7  |-  X  = 
U. J
2 eqid 2435 . . . . . . 7  |-  U. K  =  U. K
31, 2cnf 17302 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
4 ffun 5585 . . . . . 6  |-  ( F : X --> U. K  ->  Fun  F )
5 funres 5484 . . . . . 6  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
63, 4, 53syl 19 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  Fun  ( F  |`  A ) )
76adantr 452 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  Fun  ( F  |`  A ) )
8 simpr 448 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  C_  X )
93adantr 452 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  F : X --> U. K
)
10 fdm 5587 . . . . . . 7  |-  ( F : X --> U. K  ->  dom  F  =  X )
119, 10syl 16 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  dom  F  =  X )
128, 11sseqtr4d 3377 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  C_  dom  F )
13 ssdmres 5160 . . . . 5  |-  ( A 
C_  dom  F  <->  dom  ( F  |`  A )  =  A )
1412, 13sylib 189 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  dom  ( F  |`  A )  =  A )
157, 14jca 519 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
16 resss 5162 . . . . 5  |-  ( F  |`  A )  C_  F
17 rnss 5090 . . . . 5  |-  ( ( F  |`  A )  C_  F  ->  ran  ( F  |`  A )  C_  ran  F )
1816, 17ax-mp 8 . . . 4  |-  ran  ( F  |`  A )  C_  ran  F
19 frn 5589 . . . . 5  |-  ( F : X --> U. K  ->  ran  F  C_  U. K
)
209, 19syl 16 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  ran  F  C_  U. K )
2118, 20syl5ss 3351 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  ran  ( F  |`  A ) 
C_  U. K )
22 df-f 5450 . . . 4  |-  ( ( F  |`  A ) : A --> U. K  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  C_  U. K ) )
23 df-fn 5449 . . . . 5  |-  ( ( F  |`  A )  Fn  A  <->  ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
2423anbi1i 677 . . . 4  |-  ( ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A ) 
C_  U. K )  <->  ( ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A )  /\  ran  ( F  |`  A )  C_  U. K ) )
2522, 24bitri 241 . . 3  |-  ( ( F  |`  A ) : A --> U. K  <->  ( ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A )  /\  ran  ( F  |`  A )  C_  U. K ) )
2615, 21, 25sylanbrc 646 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A ) : A --> U. K
)
27 cnvresima 5351 . . . 4  |-  ( `' ( F  |`  A )
" o )  =  ( ( `' F " o )  i^i  A
)
28 cntop1 17296 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2928adantr 452 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  J  e.  Top )
3029adantr 452 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  J  e.  Top )
311topopn 16971 . . . . . . . 8  |-  ( J  e.  Top  ->  X  e.  J )
32 ssexg 4341 . . . . . . . . 9  |-  ( ( A  C_  X  /\  X  e.  J )  ->  A  e.  _V )
3332ancoms 440 . . . . . . . 8  |-  ( ( X  e.  J  /\  A  C_  X )  ->  A  e.  _V )
3431, 33sylan 458 . . . . . . 7  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  e.  _V )
3528, 34sylan 458 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  e.  _V )
3635adantr 452 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  A  e.  _V )
37 cnima 17321 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  o  e.  K )  ->  ( `' F "
o )  e.  J
)
3837adantlr 696 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  ( `' F " o )  e.  J )
39 elrestr 13648 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V  /\  ( `' F " o )  e.  J )  -> 
( ( `' F " o )  i^i  A
)  e.  ( Jt  A ) )
4030, 36, 38, 39syl3anc 1184 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  (
( `' F "
o )  i^i  A
)  e.  ( Jt  A ) )
4127, 40syl5eqel 2519 . . 3  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
4241ralrimiva 2781 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A. o  e.  K  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
431toptopon 16990 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
4428, 43sylib 189 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  (TopOn `  X )
)
45 resttopon 17217 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )
4644, 45sylan 458 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( Jt  A )  e.  (TopOn `  A ) )
47 cntop2 17297 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
4847adantr 452 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  K  e.  Top )
492toptopon 16990 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
5048, 49sylib 189 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  K  e.  (TopOn `  U. K ) )
51 iscn 17291 . . 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 ) ) ) )
5246, 50, 51syl2anc 643 . 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 ) ) ) )
5326, 42, 52mpbir2and 889 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    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    i^i cin 3311    C_ wss 3312   U.cuni 4007   `'ccnv 4869   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   ↾t crest 13640   Topctop 16950  TopOnctopon 16951    Cn ccn 17280
This theorem is referenced by:  resthauslem  17419  imacmp  17452  conima  17480  kgencn2  17581  kgencn3  17582  xkopjcn  17680  cnmpt1res  17700  cnmpt2res  17701  qtoprest  17741  hmeores  17795  ftalem3  20849  rmulccn  24306  raddcn  24307  xrge0mulc1cn  24319  rrhre  24379  cvmliftmolem1  24960  cvmlift2lem9a  24982  cvmlift2lem9  24990  areacirclem4  26284  ivthALT  26329  cnres2  26463  stoweidlem28  27744
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-fin 7105  df-fi 7408  df-rest 13642  df-topgen 13659  df-top 16955  df-bases 16957  df-topon 16958  df-cn 17283
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