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Theorem funcres2 15389
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 15353 . . 3  |-  Rel  ( C  Func  ( D  |`cat  R
) )
21a1i 11 . 2  |-  ( R  e.  (Subcat `  D
)  ->  Rel  ( C 
Func  ( D  |`cat  R
) ) )
3 simpr 459 . . . . 5  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f ( C  Func  ( D  |`cat  R
) ) g )
4 eqid 2454 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
5 eqid 2454 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
6 simpl 455 . . . . . 6  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  R  e.  (Subcat `  D ) )
7 eqidd 2455 . . . . . . 7  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  =  dom  dom  R )
86, 7subcfn 15332 . . . . . 6  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  R  Fn  ( dom  dom  R  X.  dom  dom  R ) )
9 eqid 2454 . . . . . . . 8  |-  ( Base `  ( D  |`cat  R )
)  =  ( Base `  ( D  |`cat  R )
)
104, 9, 3funcf1 15357 . . . . . . 7  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  f :
( Base `  C ) --> ( Base `  ( D  |`cat  R ) ) )
11 eqid 2454 . . . . . . . . 9  |-  ( D  |`cat 
R )  =  ( D  |`cat  R )
12 eqid 2454 . . . . . . . . 9  |-  ( Base `  D )  =  (
Base `  D )
13 subcrcl 15307 . . . . . . . . . 10  |-  ( R  e.  (Subcat `  D
)  ->  D  e.  Cat )
1413adantr 463 . . . . . . . . 9  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  D  e.  Cat )
156, 8, 12subcss1 15333 . . . . . . . . 9  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  C_  ( Base `  D
) )
1611, 12, 14, 8, 15rescbas 15320 . . . . . . . 8  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  dom  dom  R  =  ( Base `  ( D  |`cat  R ) ) )
1716feq3d 5701 . . . . . . 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 2454 . . . . . . . 8  |-  ( Hom  `  ( D  |`cat  R )
)  =  ( Hom  `  ( D  |`cat  R )
)
20 simplr 753 . . . . . . . 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 754 . . . . . . . 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 755 . . . . . . . 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 15359 . . . . . . 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 15321 . . . . . . . . . 10  |-  ( ( R  e.  (Subcat `  D )  /\  f
( C  Func  ( D  |`cat  R ) ) g )  ->  R  =  ( Hom  `  ( D  |`cat  R ) ) )
2524adantr 463 . . . . . . . . 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 6287 . . . . . . . 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 5701 . . . . . . 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 15388 . . . . 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 432 . . 3  |-  ( R  e.  (Subcat `  D
)  ->  ( f
( C  Func  ( D  |`cat  R ) ) g  ->  f ( C 
Func  D ) g ) )
32 df-br 4440 . . 3  |-  ( f ( C  Func  ( D  |`cat  R ) ) g  <->  <. f ,  g >.  e.  ( C  Func  ( D  |`cat  R ) ) )
33 df-br 4440 . . 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 5083 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 367    = wceq 1398    e. wcel 1823    C_ wss 3461   <.cop 4022   class class class wbr 4439   dom cdm 4988   Rel wrel 4993   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14719   Hom chom 14798   Catccat 15156    |`cat cresc 15299  Subcatcsubc 15300    Func cfunc 15345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-hom 14811  df-cco 14812  df-cat 15160  df-cid 15161  df-homf 15162  df-ssc 15301  df-resc 15302  df-subc 15303  df-func 15349
This theorem is referenced by:  fthres2  15423  ressffth  15429  funcsetcres2  15574
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