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Theorem funcres2 14907
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 14871 . . 3  |-  Rel  ( C  Func  ( D  |`cat  R
) )
21a1i 11 . 2  |-  ( R  e.  (Subcat `  D
)  ->  Rel  ( C 
Func  ( D  |`cat  R
) ) )
3 simpr 461 . . . . 5  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f ( C  Func  ( D  |`cat  R
) ) g )
4 eqid 2451 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
5 eqid 2451 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
6 simpl 457 . . . . . 6  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  R  e.  (Subcat `  D ) )
7 eqidd 2452 . . . . . . 7  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  =  dom  dom  R )
86, 7subcfn 14850 . . . . . 6  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  R  Fn  ( dom  dom  R  X.  dom  dom  R ) )
9 eqid 2451 . . . . . . . 8  |-  ( Base `  ( D  |`cat  R )
)  =  ( Base `  ( D  |`cat  R )
)
104, 9, 3funcf1 14875 . . . . . . 7  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f :
( Base `  C ) --> ( Base `  ( D  |`cat  R ) ) )
11 eqid 2451 . . . . . . . . 9  |-  ( D  |`cat 
R )  =  ( D  |`cat  R )
12 eqid 2451 . . . . . . . . 9  |-  ( Base `  D )  =  (
Base `  D )
13 subcrcl 14828 . . . . . . . . . 10  |-  ( R  e.  (Subcat `  D
)  ->  D  e.  Cat )
1413adantr 465 . . . . . . . . 9  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  D  e.  Cat )
156, 8, 12subcss1 14851 . . . . . . . . 9  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  C_  ( Base `  D
) )
1611, 12, 14, 8, 15rescbas 14841 . . . . . . . 8  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  =  ( Base `  ( D  |`cat  R ) ) )
17 feq3 5639 . . . . . . . 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 232 . . . . . 6  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f :
( Base `  C ) --> dom  dom  R )
20 eqid 2451 . . . . . . . 8  |-  ( Hom  `  ( D  |`cat  R )
)  =  ( Hom  `  ( D  |`cat  R )
)
21 simplr 754 . . . . . . . 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 755 . . . . . . . 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 756 . . . . . . . 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 14877 . . . . . . 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 14842 . . . . . . . . . 10  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  R  =  ( Hom  `  ( D  |`cat  R ) ) )
2625adantr 465 . . . . . . . . 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 6204 . . . . . . . 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 5639 . . . . . . . 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 232 . . . . . 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 14906 . . . . 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 232 . . . 4  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f ( C  Func  D ) g )
3332ex 434 . . 3  |-  ( R  e.  (Subcat `  D
)  ->  ( f
( C  Func  ( D  |`cat  R ) ) g  ->  f ( C 
Func  D ) g ) )
34 df-br 4388 . . 3  |-  ( f ( C  Func  ( D  |`cat  R ) ) g  <->  <. f ,  g >.  e.  ( C  Func  ( D  |`cat  R ) ) )
35 df-br 4388 . . 3  |-  ( f ( C  Func  D
) g  <->  <. f ,  g >.  e.  ( C  Func  D ) )
3633, 34, 353imtr3g 269 . 2  |-  ( R  e.  (Subcat `  D
)  ->  ( <. f ,  g >.  e.  ( C  Func  ( D  |`cat  R ) )  ->  <. f ,  g >.  e.  ( C  Func  D )
) )
372, 36relssdv 5027 1  |-  ( R  e.  (Subcat `  D
)  ->  ( C  Func  ( D  |`cat  R )
)  C_  ( C  Func  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3423   <.cop 3978   class class class wbr 4387   dom cdm 4935   Rel wrel 4940   -->wf 5509   ` cfv 5513  (class class class)co 6187   Basecbs 14273   Hom chom 14348   Catccat 14701    |`cat cresc 14820  Subcatcsubc 14821    Func cfunc 14863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-er 7198  df-map 7313  df-pm 7314  df-ixp 7361  df-en 7408  df-dom 7409  df-sdom 7410  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-2 10478  df-3 10479  df-4 10480  df-5 10481  df-6 10482  df-7 10483  df-8 10484  df-9 10485  df-10 10486  df-n0 10678  df-z 10745  df-dec 10854  df-ndx 14276  df-slot 14277  df-base 14278  df-sets 14279  df-ress 14280  df-hom 14361  df-cco 14362  df-cat 14705  df-cid 14706  df-homf 14707  df-ssc 14822  df-resc 14823  df-subc 14824  df-func 14867
This theorem is referenced by:  fthres2  14941  ressffth  14947  funcsetcres2  15060
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