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Theorem funcres2 13788
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 13752 . . 3  |-  Rel  ( C  Func  ( D  |`cat  R
) )
21a1i 10 . 2  |-  ( R  e.  (Subcat `  D
)  ->  Rel  ( C 
Func  ( D  |`cat  R
) ) )
3 simpr 447 . . . . 5  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f ( C  Func  ( D  |`cat  R
) ) g )
4 eqid 2296 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
5 eqid 2296 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
6 simpl 443 . . . . . 6  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  R  e.  (Subcat `  D ) )
7 eqidd 2297 . . . . . . 7  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  =  dom  dom  R )
86, 7subcfn 13731 . . . . . 6  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  R  Fn  ( dom  dom  R  X.  dom  dom  R ) )
9 eqid 2296 . . . . . . . 8  |-  ( Base `  ( D  |`cat  R )
)  =  ( Base `  ( D  |`cat  R )
)
104, 9, 3funcf1 13756 . . . . . . 7  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f :
( Base `  C ) --> ( Base `  ( D  |`cat  R ) ) )
11 eqid 2296 . . . . . . . . 9  |-  ( D  |`cat 
R )  =  ( D  |`cat  R )
12 eqid 2296 . . . . . . . . 9  |-  ( Base `  D )  =  (
Base `  D )
13 subcrcl 13709 . . . . . . . . . 10  |-  ( R  e.  (Subcat `  D
)  ->  D  e.  Cat )
1413adantr 451 . . . . . . . . 9  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  D  e.  Cat )
156, 8, 12subcss1 13732 . . . . . . . . 9  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  C_  ( Base `  D
) )
1611, 12, 14, 8, 15rescbas 13722 . . . . . . . 8  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  =  ( Base `  ( D  |`cat  R ) ) )
17 feq3 5393 . . . . . . . 8  |-  ( dom 
dom  R  =  ( Base `  ( D  |`cat  R
) )  ->  (
f : ( Base `  C ) --> dom  dom  R  <-> 
f : ( Base `  C ) --> ( Base `  ( D  |`cat  R )
) ) )
1816, 17syl 15 . . . . . . 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 223 . . . . . 6  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f :
( Base `  C ) --> dom  dom  R )
20 eqid 2296 . . . . . . . 8  |-  (  Hom  `  ( D  |`cat  R )
)  =  (  Hom  `  ( D  |`cat  R )
)
21 simplr 731 . . . . . . . 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 732 . . . . . . . 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 733 . . . . . . . 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 13758 . . . . . . 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 13723 . . . . . . . . . 10  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  R  =  (  Hom  `  ( D  |`cat  R ) ) )
2625adantr 451 . . . . . . . . 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 5891 . . . . . . . 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 5393 . . . . . . . 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 15 . . . . . . 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 223 . . . . . 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 13787 . . . . 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 223 . . . 4  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f ( C  Func  D ) g )
3332ex 423 . . 3  |-  ( R  e.  (Subcat `  D
)  ->  ( f
( C  Func  ( D  |`cat  R ) ) g  ->  f ( C 
Func  D ) g ) )
34 df-br 4040 . . 3  |-  ( f ( C  Func  ( D  |`cat  R ) ) g  <->  <. f ,  g >.  e.  ( C  Func  ( D  |`cat  R ) ) )
35 df-br 4040 . . 3  |-  ( f ( C  Func  D
) g  <->  <. f ,  g >.  e.  ( C  Func  D ) )
3633, 34, 353imtr3g 260 . 2  |-  ( R  e.  (Subcat `  D
)  ->  ( <. f ,  g >.  e.  ( C  Func  ( D  |`cat  R ) )  ->  <. f ,  g >.  e.  ( C  Func  D )
) )
372, 36relssdv 4795 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   <.cop 3656   class class class wbr 4039   dom cdm 4705   Rel wrel 4710   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235   Catccat 13582    |`cat cresc 13701  Subcatcsubc 13702    Func cfunc 13744
This theorem is referenced by:  fthres2  13822  ressffth  13828  funcsetcres2  13941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-hom 13248  df-cco 13249  df-cat 13586  df-cid 13587  df-homf 13588  df-ssc 13703  df-resc 13704  df-subc 13705  df-func 13748
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