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Theorem funcres2 15136
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 15100 . . 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 2441 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
5 eqid 2441 . . . . . 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 2442 . . . . . . 7  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  =  dom  dom  R )
86, 7subcfn 15079 . . . . . 6  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  R  Fn  ( dom  dom  R  X.  dom  dom  R ) )
9 eqid 2441 . . . . . . . 8  |-  ( Base `  ( D  |`cat  R )
)  =  ( Base `  ( D  |`cat  R )
)
104, 9, 3funcf1 15104 . . . . . . 7  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f :
( Base `  C ) --> ( Base `  ( D  |`cat  R ) ) )
11 eqid 2441 . . . . . . . . 9  |-  ( D  |`cat 
R )  =  ( D  |`cat  R )
12 eqid 2441 . . . . . . . . 9  |-  ( Base `  D )  =  (
Base `  D )
13 subcrcl 15057 . . . . . . . . . 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 15080 . . . . . . . . 9  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  C_  ( Base `  D
) )
1611, 12, 14, 8, 15rescbas 15070 . . . . . . . 8  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  =  ( Base `  ( D  |`cat  R ) ) )
1716feq3d 5705 . . . . . . 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 ) ) ) )
1810, 17mpbird 232 . . . . . 6  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f :
( Base `  C ) --> dom  dom  R )
19 eqid 2441 . . . . . . . 8  |-  ( Hom  `  ( D  |`cat  R )
)  =  ( Hom  `  ( D  |`cat  R )
)
20 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 )
21 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
) )
22 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
) )
234, 5, 19, 20, 21, 22funcf2 15106 . . . . . . 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 ) ) )
2411, 12, 14, 8, 15reschom 15071 . . . . . . . . . 10  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  R  =  ( Hom  `  ( D  |`cat  R ) ) )
2524adantr 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 ) ) )
2625oveqd 6294 . . . . . . . 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 ) ) )
2726feq3d 5705 . . . . . . 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 ) ) ) )
2823, 27mpbird 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 ) ) )
294, 5, 6, 8, 18, 28funcres2b 15135 . . . . 5  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  ( f
( C  Func  D
) g  <->  f ( C  Func  ( D  |`cat  R
) ) g ) )
303, 29mpbird 232 . . . 4  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f ( C  Func  D ) g )
3130ex 434 . . 3  |-  ( R  e.  (Subcat `  D
)  ->  ( f
( C  Func  ( D  |`cat  R ) ) g  ->  f ( C 
Func  D ) g ) )
32 df-br 4434 . . 3  |-  ( f ( C  Func  ( D  |`cat  R ) ) g  <->  <. f ,  g >.  e.  ( C  Func  ( D  |`cat  R ) ) )
33 df-br 4434 . . 3  |-  ( f ( C  Func  D
) g  <->  <. f ,  g >.  e.  ( C  Func  D ) )
3431, 32, 333imtr3g 269 . 2  |-  ( R  e.  (Subcat `  D
)  ->  ( <. f ,  g >.  e.  ( C  Func  ( D  |`cat  R ) )  ->  <. f ,  g >.  e.  ( C  Func  D )
) )
352, 34relssdv 5081 1  |-  ( R  e.  (Subcat `  D
)  ->  ( C  Func  ( D  |`cat  R )
)  C_  ( C  Func  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802    C_ wss 3458   <.cop 4016   class class class wbr 4433   dom cdm 4985   Rel wrel 4990   -->wf 5570   ` cfv 5574  (class class class)co 6277   Basecbs 14504   Hom chom 14580   Catccat 14933    |`cat cresc 15049  Subcatcsubc 15050    Func cfunc 15092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-map 7420  df-pm 7421  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-hom 14593  df-cco 14594  df-cat 14937  df-cid 14938  df-homf 14939  df-ssc 15051  df-resc 15052  df-subc 15053  df-func 15096
This theorem is referenced by:  fthres2  15170  ressffth  15176  funcsetcres2  15289
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