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Theorem cnrest 17351
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 2438 . . . . . . 7  |-  U. K  =  U. K
31, 2cnf 17312 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
4 ffun 5595 . . . . . 6  |-  ( F : X --> U. K  ->  Fun  F )
5 funres 5494 . . . . . 6  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
63, 4, 53syl 19 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  Fun  ( F  |`  A ) )
76adantr 453 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  Fun  ( F  |`  A ) )
8 simpr 449 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  C_  X )
93adantr 453 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  F : X --> U. K
)
10 fdm 5597 . . . . . . 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 3387 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  C_  dom  F )
13 ssdmres 5170 . . . . 5  |-  ( A 
C_  dom  F  <->  dom  ( F  |`  A )  =  A )
1412, 13sylib 190 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  dom  ( F  |`  A )  =  A )
157, 14jca 520 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
16 resss 5172 . . . . 5  |-  ( F  |`  A )  C_  F
17 rnss 5100 . . . . 5  |-  ( ( F  |`  A )  C_  F  ->  ran  ( F  |`  A )  C_  ran  F )
1816, 17ax-mp 8 . . . 4  |-  ran  ( F  |`  A )  C_  ran  F
19 frn 5599 . . . . 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 3361 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  ran  ( F  |`  A ) 
C_  U. K )
22 df-f 5460 . . . 4  |-  ( ( F  |`  A ) : A --> U. K  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  C_  U. K ) )
23 df-fn 5459 . . . . 5  |-  ( ( F  |`  A )  Fn  A  <->  ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
2423anbi1i 678 . . . 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 242 . . 3  |-  ( ( F  |`  A ) : A --> U. K  <->  ( ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A )  /\  ran  ( F  |`  A )  C_  U. K ) )
2615, 21, 25sylanbrc 647 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A ) : A --> U. K
)
27 cnvresima 5361 . . . 4  |-  ( `' ( F  |`  A )
" o )  =  ( ( `' F " o )  i^i  A
)
28 cntop1 17306 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
2928adantr 453 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  J  e.  Top )
3029adantr 453 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  J  e.  Top )
311topopn 16981 . . . . . . . 8  |-  ( J  e.  Top  ->  X  e.  J )
32 ssexg 4351 . . . . . . . . 9  |-  ( ( A  C_  X  /\  X  e.  J )  ->  A  e.  _V )
3332ancoms 441 . . . . . . . 8  |-  ( ( X  e.  J  /\  A  C_  X )  ->  A  e.  _V )
3431, 33sylan 459 . . . . . . 7  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  e.  _V )
3528, 34sylan 459 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A  e.  _V )
3635adantr 453 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  A  e.  _V )
37 cnima 17331 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  o  e.  K )  ->  ( `' F "
o )  e.  J
)
3837adantlr 697 . . . . 5  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  ( `' F " o )  e.  J )
39 elrestr 13658 . . . . 5  |-  ( ( J  e.  Top  /\  A  e.  _V  /\  ( `' F " o )  e.  J )  -> 
( ( `' F " o )  i^i  A
)  e.  ( Jt  A ) )
4030, 36, 38, 39syl3anc 1185 . . . 4  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  (
( `' F "
o )  i^i  A
)  e.  ( Jt  A ) )
4127, 40syl5eqel 2522 . . 3  |-  ( ( ( F  e.  ( J  Cn  K )  /\  A  C_  X
)  /\  o  e.  K )  ->  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
4241ralrimiva 2791 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  A. o  e.  K  ( `' ( F  |`  A ) " o
)  e.  ( Jt  A ) )
431toptopon 17000 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
4428, 43sylib 190 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  (TopOn `  X )
)
45 resttopon 17227 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )
4644, 45sylan 459 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( Jt  A )  e.  (TopOn `  A ) )
47 cntop2 17307 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
4847adantr 453 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  K  e.  Top )
492toptopon 17000 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
5048, 49sylib 190 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  ->  K  e.  (TopOn `  U. K ) )
51 iscn 17301 . . 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 644 . 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 890 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    i^i cin 3321    C_ wss 3322   U.cuni 4017   `'ccnv 4879   dom cdm 4880   ran crn 4881    |` cres 4882   "cima 4883   Fun wfun 5450    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   ↾t crest 13650   Topctop 16960  TopOnctopon 16961    Cn ccn 17290
This theorem is referenced by:  resthauslem  17429  imacmp  17462  conima  17490  kgencn2  17591  kgencn3  17592  xkopjcn  17690  cnmpt1res  17710  cnmpt2res  17711  qtoprest  17751  hmeores  17805  ftalem3  20859  rmulccn  24316  raddcn  24317  xrge0mulc1cn  24329  rrhre  24389  cvmliftmolem1  24970  cvmlift2lem9a  24992  cvmlift2lem9  25000  areacirclem2  26295  ivthALT  26340  cnres2  26474  stoweidlem28  27755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-recs 6635  df-rdg 6670  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-fin 7115  df-fi 7418  df-rest 13652  df-topgen 13669  df-top 16965  df-bases 16967  df-topon 16968  df-cn 17293
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