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Theorem cnrest 17119
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 2358 . . . . . . 7  |-  U. K  =  U. K
31, 2cnf 17082 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
4 ffun 5474 . . . . . 6  |-  ( F : X --> U. K  ->  Fun  F )
5 funres 5375 . . . . . 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 5476 . . . . . . 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 3291 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  C_  dom  F )
13 ssdmres 5059 . . . . 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 5061 . . . . 5  |-  ( F  |`  A )  C_  F
17 rnss 4989 . . . . 5  |-  ( ( F  |`  A )  C_  F  ->  ran  ( F  |`  A )  C_  ran  F )
1816, 17ax-mp 8 . . . 4  |-  ran  ( F  |`  A )  C_  ran  F
19 frn 5478 . . . . 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 3266 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  ran  ( F  |`  A ) 
C_  U. K )
22 df-f 5341 . . . 4  |-  ( ( F  |`  A ) : A --> U. K  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  C_  U. K ) )
23 df-fn 5340 . . . . 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 5244 . . . 4  |-  ( `' ( F  |`  A )
" o )  =  ( ( `' F " o )  i^i  A
)
28 cntop1 17076 . . . . . . 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 16758 . . . . . . . 8  |-  ( J  e.  Top  ->  X  e.  J )
32 ssexg 4241 . . . . . . . . 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 17100 . . . . . 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 13432 . . . . 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 2442 . . 3  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
4241ralrimiva 2702 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A. o  e.  K  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
431toptopon 16777 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
4428, 43sylib 188 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  (TopOn `  X )
)
45 resttopon 16998 . . . 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 17077 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
4847adantr 451 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  K  e.  Top )
492toptopon 16777 . . . 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 17071 . . 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 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864    i^i cin 3227    C_ wss 3228   U.cuni 3908   `'ccnv 4770   dom cdm 4771   ran crn 4772    |` cres 4773   "cima 4774   Fun wfun 5331    Fn wfn 5332   -->wf 5333   ` cfv 5337  (class class class)co 5945   ↾t crest 13424   Topctop 16737  TopOnctopon 16738    Cn ccn 17060
This theorem is referenced by:  resthauslem  17197  imacmp  17230  conima  17257  kgencn2  17358  kgencn3  17359  xkopjcn  17456  cnmpt1res  17476  cnmpt2res  17477  qtoprest  17514  hmeores  17568  ftalem3  20424  rmulccn  23470  raddcn  23471  xrge0mulc1cn  23483  cvmliftmolem1  24216  cvmlift2lem9a  24238  cvmlift2lem9  24246  areacirclem4  25519  ivthALT  25582  cnres2  25806  stoweidlem28  27100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-recs 6475  df-rdg 6510  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-fin 6955  df-fi 7255  df-rest 13426  df-topgen 13443  df-top 16742  df-bases 16744  df-topon 16745  df-cn 17063
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