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Theorem fullfunc 13780
Description: A full functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
Assertion
Ref Expression
fullfunc  |-  ( C Full 
D )  C_  ( C  Func  D )

Proof of Theorem fullfunc
Dummy variables  c 
d  f  g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5865 . . . 4  |-  ( c  =  C  ->  (
c Full  d )  =  ( C Full  d ) )
2 oveq1 5865 . . . 4  |-  ( c  =  C  ->  (
c  Func  d )  =  ( C  Func  d ) )
31, 2sseq12d 3207 . . 3  |-  ( c  =  C  ->  (
( c Full  d ) 
C_  ( c  Func  d )  <->  ( C Full  d
)  C_  ( C  Func  d ) ) )
4 oveq2 5866 . . . 4  |-  ( d  =  D  ->  ( C Full  d )  =  ( C Full  D ) )
5 oveq2 5866 . . . 4  |-  ( d  =  D  ->  ( C  Func  d )  =  ( C  Func  D
) )
64, 5sseq12d 3207 . . 3  |-  ( d  =  D  ->  (
( C Full  d )  C_  ( C  Func  d
)  <->  ( C Full  D
)  C_  ( C  Func  D ) ) )
7 ovex 5883 . . . . . 6  |-  ( c 
Func  d )  e. 
_V
8 simpl 443 . . . . . . . 8  |-  ( ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  d ) ( f `
 y ) ) )  ->  f (
c  Func  d )
g )
98ssopab2i 4292 . . . . . . 7  |-  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  d ) ( f `
 y ) ) ) }  C_  { <. f ,  g >.  |  f ( c  Func  d
) g }
10 opabss 4080 . . . . . . 7  |-  { <. f ,  g >.  |  f ( c  Func  d
) g }  C_  ( c  Func  d
)
119, 10sstri 3188 . . . . . 6  |-  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  d ) ( f `
 y ) ) ) }  C_  (
c  Func  d )
127, 11ssexi 4159 . . . . 5  |-  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  d ) ( f `
 y ) ) ) }  e.  _V
13 df-full 13778 . . . . . 6  |- Full  =  ( c  e.  Cat , 
d  e.  Cat  |->  {
<. f ,  g >.  |  ( f ( c  Func  d )
g  /\  A. x  e.  ( Base `  c
) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  d ) ( f `
 y ) ) ) } )
1413ovmpt4g 5970 . . . . 5  |-  ( ( c  e.  Cat  /\  d  e.  Cat  /\  { <. f ,  g >.  |  ( f ( c  Func  d )
g  /\  A. x  e.  ( Base `  c
) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  d ) ( f `
 y ) ) ) }  e.  _V )  ->  ( c Full  d
)  =  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  d ) ( f `
 y ) ) ) } )
1512, 14mp3an3 1266 . . . 4  |-  ( ( c  e.  Cat  /\  d  e.  Cat )  ->  ( c Full  d )  =  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) (  Hom  `  d ) ( f `
 y ) ) ) } )
1615, 11syl6eqss 3228 . . 3  |-  ( ( c  e.  Cat  /\  d  e.  Cat )  ->  ( c Full  d ) 
C_  ( c  Func  d ) )
173, 6, 16vtocl2ga 2851 . 2  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Full  D ) 
C_  ( C  Func  D ) )
1813, 12fnmpt2i 6193 . . . . 5  |- Full  Fn  ( Cat  X.  Cat )
19 fndm 5343 . . . . 5  |-  ( Full  Fn  ( Cat  X.  Cat )  ->  dom Full  =  ( Cat 
X.  Cat ) )
2018, 19ax-mp 8 . . . 4  |-  dom Full  =  ( Cat  X.  Cat )
2120ndmov 6004 . . 3  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Full  D )  =  (/) )
22 0ss 3483 . . 3  |-  (/)  C_  ( C  Func  D )
2321, 22syl6eqss 3228 . 2  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Full  D ) 
C_  ( C  Func  D ) )
2417, 23pm2.61i 156 1  |-  ( C Full 
D )  C_  ( C  Func  D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   (/)c0 3455   class class class wbr 4023   {copab 4076    X. cxp 4687   dom cdm 4689   ran crn 4690    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219   Catccat 13566    Func cfunc 13728   Full cful 13776
This theorem is referenced by:  relfull  13782  isfull  13784  fulloppc  13796  cofull  13808  catcisolem  13938  catciso  13939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-full 13778
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