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Theorem cnrest 17271
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 2387 . . . . . . 7  |-  U. K  =  U. K
31, 2cnf 17232 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
4 ffun 5533 . . . . . 6  |-  ( F : X --> U. K  ->  Fun  F )
5 funres 5432 . . . . . 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 5535 . . . . . . 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 3328 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  C_  dom  F )
13 ssdmres 5108 . . . . 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 5110 . . . . 5  |-  ( F  |`  A )  C_  F
17 rnss 5038 . . . . 5  |-  ( ( F  |`  A )  C_  F  ->  ran  ( F  |`  A )  C_  ran  F )
1816, 17ax-mp 8 . . . 4  |-  ran  ( F  |`  A )  C_  ran  F
19 frn 5537 . . . . 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 3302 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  ran  ( F  |`  A ) 
C_  U. K )
22 df-f 5398 . . . 4  |-  ( ( F  |`  A ) : A --> U. K  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  C_  U. K ) )
23 df-fn 5397 . . . . 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 5299 . . . 4  |-  ( `' ( F  |`  A )
" o )  =  ( ( `' F " o )  i^i  A
)
28 cntop1 17226 . . . . . . 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 16902 . . . . . . . 8  |-  ( J  e.  Top  ->  X  e.  J )
32 ssexg 4290 . . . . . . . . 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 17251 . . . . . 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 13583 . . . . 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 2471 . . 3  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
4241ralrimiva 2732 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A. o  e.  K  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
431toptopon 16921 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
4428, 43sylib 189 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  (TopOn `  X )
)
45 resttopon 17147 . . . 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 17227 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
4847adantr 452 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  K  e.  Top )
492toptopon 16921 . . . 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 17221 . . 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 1649    e. wcel 1717   A.wral 2649   _Vcvv 2899    i^i cin 3262    C_ wss 3263   U.cuni 3957   `'ccnv 4817   dom cdm 4818   ran crn 4819    |` cres 4820   "cima 4821   Fun wfun 5388    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   ↾t crest 13575   Topctop 16881  TopOnctopon 16882    Cn ccn 17210
This theorem is referenced by:  resthauslem  17349  imacmp  17382  conima  17409  kgencn2  17510  kgencn3  17511  xkopjcn  17609  cnmpt1res  17629  cnmpt2res  17630  qtoprest  17670  hmeores  17724  ftalem3  20724  rmulccn  24118  raddcn  24119  xrge0mulc1cn  24131  rrhre  24183  cvmliftmolem1  24747  cvmlift2lem9a  24769  cvmlift2lem9  24777  areacirclem4  25984  ivthALT  26029  cnres2  26163  stoweidlem28  27445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-recs 6569  df-rdg 6604  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-fin 7049  df-fi 7351  df-rest 13577  df-topgen 13594  df-top 16886  df-bases 16888  df-topon 16889  df-cn 17213
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