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Theorem cnrest 17013
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 2283 . . . . . . 7  |-  U. K  =  U. K
31, 2cnf 16976 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
4 ffun 5391 . . . . . 6  |-  ( F : X --> U. K  ->  Fun  F )
5 funres 5293 . . . . . 6  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
63, 4, 53syl 18 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  Fun  ( F  |`  A ) )
76adantr 451 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  Fun  ( F  |`  A ) )
8 simpr 447 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  C_  X )
93adantr 451 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  F : X --> U. K
)
10 fdm 5393 . . . . . . 7  |-  ( F : X --> U. K  ->  dom  F  =  X )
119, 10syl 15 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  dom  F  =  X )
128, 11sseqtr4d 3215 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  C_  dom  F )
13 ssdmres 4977 . . . . 5  |-  ( A 
C_  dom  F  <->  dom  ( F  |`  A )  =  A )
1412, 13sylib 188 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  dom  ( F  |`  A )  =  A )
157, 14jca 518 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
16 resss 4979 . . . . 5  |-  ( F  |`  A )  C_  F
17 rnss 4907 . . . . 5  |-  ( ( F  |`  A )  C_  F  ->  ran  ( F  |`  A )  C_  ran  F )
1816, 17ax-mp 8 . . . 4  |-  ran  ( F  |`  A )  C_  ran  F
19 frn 5395 . . . . 5  |-  ( F : X --> U. K  ->  ran  F  C_  U. K
)
209, 19syl 15 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  ran  F  C_  U. K )
2118, 20syl5ss 3190 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  ran  ( F  |`  A ) 
C_  U. K )
22 df-f 5259 . . . 4  |-  ( ( F  |`  A ) : A --> U. K  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  C_  U. K ) )
23 df-fn 5258 . . . . 5  |-  ( ( F  |`  A )  Fn  A  <->  ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
2423anbi1i 676 . . . 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 240 . . 3  |-  ( ( F  |`  A ) : A --> U. K  <->  ( ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A )  /\  ran  ( F  |`  A )  C_  U. K ) )
2615, 21, 25sylanbrc 645 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A ) : A --> U. K
)
27 cnvresima 5162 . . . 4  |-  ( `' ( F  |`  A )
" o )  =  ( ( `' F " o )  i^i  A
)
28 cntop1 16970 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2928adantr 451 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  J  e.  Top )
3029adantr 451 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  J  e.  Top )
311topopn 16652 . . . . . . . 8  |-  ( J  e.  Top  ->  X  e.  J )
32 ssexg 4160 . . . . . . . . 9  |-  ( ( A  C_  X  /\  X  e.  J )  ->  A  e.  _V )
3332ancoms 439 . . . . . . . 8  |-  ( ( X  e.  J  /\  A  C_  X )  ->  A  e.  _V )
3431, 33sylan 457 . . . . . . 7  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  e.  _V )
3528, 34sylan 457 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  e.  _V )
3635adantr 451 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  A  e.  _V )
37 cnima 16994 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  o  e.  K )  ->  ( `' F "
o )  e.  J
)
3837adantlr 695 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  ( `' F " o )  e.  J )
39 elrestr 13333 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V  /\  ( `' F " o )  e.  J )  -> 
( ( `' F " o )  i^i  A
)  e.  ( Jt  A ) )
4030, 36, 38, 39syl3anc 1182 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  (
( `' F "
o )  i^i  A
)  e.  ( Jt  A ) )
4127, 40syl5eqel 2367 . . 3  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
4241ralrimiva 2626 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A. o  e.  K  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
431toptopon 16671 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
4428, 43sylib 188 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  (TopOn `  X )
)
45 resttopon 16892 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )
4644, 45sylan 457 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( Jt  A )  e.  (TopOn `  A ) )
47 cntop2 16971 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
4847adantr 451 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  K  e.  Top )
492toptopon 16671 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
5048, 49sylib 188 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  K  e.  (TopOn `  U. K ) )
51 iscn 16965 . . 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 642 . 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 888 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   U.cuni 3827   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  resthauslem  17091  imacmp  17124  conima  17151  kgencn2  17252  kgencn3  17253  xkopjcn  17350  cnmpt1res  17370  cnmpt2res  17371  qtoprest  17408  hmeores  17462  ftalem3  20312  rmulccn  23301  raddcn  23302  xrge0mulc1cn  23323  cvmliftmolem1  23812  cvmlift2lem9a  23834  cvmlift2lem9  23842  areacirclem4  24927  ivthALT  26258  cnres2  26483  stoweidlem28  27777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957
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