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Theorem funcres2 14087
Description: A functor into a restricted category is also a functor into the whole category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
funcres2  |-  ( R  e.  (Subcat `  D
)  ->  ( C  Func  ( D  |`cat  R )
)  C_  ( C  Func  D ) )

Proof of Theorem funcres2
Dummy variables  f 
g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfunc 14051 . . 3  |-  Rel  ( C  Func  ( D  |`cat  R
) )
21a1i 11 . 2  |-  ( R  e.  (Subcat `  D
)  ->  Rel  ( C 
Func  ( D  |`cat  R
) ) )
3 simpr 448 . . . . 5  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f ( C  Func  ( D  |`cat  R
) ) g )
4 eqid 2435 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
5 eqid 2435 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
6 simpl 444 . . . . . 6  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  R  e.  (Subcat `  D ) )
7 eqidd 2436 . . . . . . 7  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  =  dom  dom  R )
86, 7subcfn 14030 . . . . . 6  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  R  Fn  ( dom  dom  R  X.  dom  dom  R ) )
9 eqid 2435 . . . . . . . 8  |-  ( Base `  ( D  |`cat  R )
)  =  ( Base `  ( D  |`cat  R )
)
104, 9, 3funcf1 14055 . . . . . . 7  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f :
( Base `  C ) --> ( Base `  ( D  |`cat  R ) ) )
11 eqid 2435 . . . . . . . . 9  |-  ( D  |`cat 
R )  =  ( D  |`cat  R )
12 eqid 2435 . . . . . . . . 9  |-  ( Base `  D )  =  (
Base `  D )
13 subcrcl 14008 . . . . . . . . . 10  |-  ( R  e.  (Subcat `  D
)  ->  D  e.  Cat )
1413adantr 452 . . . . . . . . 9  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  D  e.  Cat )
156, 8, 12subcss1 14031 . . . . . . . . 9  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  C_  ( Base `  D
) )
1611, 12, 14, 8, 15rescbas 14021 . . . . . . . 8  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  =  ( Base `  ( D  |`cat  R ) ) )
17 feq3 5570 . . . . . . . 8  |-  ( dom 
dom  R  =  ( Base `  ( D  |`cat  R
) )  ->  (
f : ( Base `  C ) --> dom  dom  R  <-> 
f : ( Base `  C ) --> ( Base `  ( D  |`cat  R )
) ) )
1816, 17syl 16 . . . . . . 7  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  ( f : ( Base `  C
) --> dom  dom  R  <->  f :
( Base `  C ) --> ( Base `  ( D  |`cat  R ) ) ) )
1910, 18mpbird 224 . . . . . 6  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f :
( Base `  C ) --> dom  dom  R )
20 eqid 2435 . . . . . . . 8  |-  (  Hom  `  ( D  |`cat  R )
)  =  (  Hom  `  ( D  |`cat  R )
)
21 simplr 732 . . . . . . . 8  |-  ( ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  f
( C  Func  ( D  |`cat  R ) ) g )
22 simprl 733 . . . . . . . 8  |-  ( ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
23 simprr 734 . . . . . . . 8  |-  ( ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
244, 5, 20, 21, 22, 23funcf2 14057 . . . . . . 7  |-  ( ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x g y ) : ( x (  Hom  `  C )
y ) --> ( ( f `  x ) (  Hom  `  ( D  |`cat  R ) ) ( f `  y ) ) )
2511, 12, 14, 8, 15reschom 14022 . . . . . . . . . 10  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  R  =  (  Hom  `  ( D  |`cat  R ) ) )
2625adantr 452 . . . . . . . . 9  |-  ( ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  R  =  (  Hom  `  ( D  |`cat  R ) ) )
2726oveqd 6090 . . . . . . . 8  |-  ( ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( f `  x
) R ( f `
 y ) )  =  ( ( f `
 x ) (  Hom  `  ( D  |`cat  R ) ) ( f `
 y ) ) )
28 feq3 5570 . . . . . . . 8  |-  ( ( ( f `  x
) R ( f `
 y ) )  =  ( ( f `
 x ) (  Hom  `  ( D  |`cat  R ) ) ( f `
 y ) )  ->  ( ( x g y ) : ( x (  Hom  `  C ) y ) --> ( ( f `  x ) R ( f `  y ) )  <->  ( x g y ) : ( x (  Hom  `  C
) y ) --> ( ( f `  x
) (  Hom  `  ( D  |`cat  R ) ) ( f `  y ) ) ) )
2927, 28syl 16 . . . . . . 7  |-  ( ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( x g y ) : ( x (  Hom  `  C
) y ) --> ( ( f `  x
) R ( f `
 y ) )  <-> 
( x g y ) : ( x (  Hom  `  C
) y ) --> ( ( f `  x
) (  Hom  `  ( D  |`cat  R ) ) ( f `  y ) ) ) )
3024, 29mpbird 224 . . . . . 6  |-  ( ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x g y ) : ( x (  Hom  `  C )
y ) --> ( ( f `  x ) R ( f `  y ) ) )
314, 5, 6, 8, 19, 30funcres2b 14086 . . . . 5  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  ( f
( C  Func  D
) g  <->  f ( C  Func  ( D  |`cat  R
) ) g ) )
323, 31mpbird 224 . . . 4  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f ( C  Func  D ) g )
3332ex 424 . . 3  |-  ( R  e.  (Subcat `  D
)  ->  ( f
( C  Func  ( D  |`cat  R ) ) g  ->  f ( C 
Func  D ) g ) )
34 df-br 4205 . . 3  |-  ( f ( C  Func  ( D  |`cat  R ) ) g  <->  <. f ,  g >.  e.  ( C  Func  ( D  |`cat  R ) ) )
35 df-br 4205 . . 3  |-  ( f ( C  Func  D
) g  <->  <. f ,  g >.  e.  ( C  Func  D ) )
3633, 34, 353imtr3g 261 . 2  |-  ( R  e.  (Subcat `  D
)  ->  ( <. f ,  g >.  e.  ( C  Func  ( D  |`cat  R ) )  ->  <. f ,  g >.  e.  ( C  Func  D )
) )
372, 36relssdv 4960 1  |-  ( R  e.  (Subcat `  D
)  ->  ( C  Func  ( D  |`cat  R )
)  C_  ( C  Func  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   <.cop 3809   class class class wbr 4204   dom cdm 4870   Rel wrel 4875   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461    Hom chom 13532   Catccat 13881    |`cat cresc 14000  Subcatcsubc 14001    Func cfunc 14043
This theorem is referenced by:  fthres2  14121  ressffth  14127  funcsetcres2  14240
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  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  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-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-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-hom 13545  df-cco 13546  df-cat 13885  df-cid 13886  df-homf 13887  df-ssc 14002  df-resc 14003  df-subc 14004  df-func 14047
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