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Theorem fullfunc 14816
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 6098 . . . 4  |-  ( c  =  C  ->  (
c Full  d )  =  ( C Full  d ) )
2 oveq1 6098 . . . 4  |-  ( c  =  C  ->  (
c  Func  d )  =  ( C  Func  d ) )
31, 2sseq12d 3385 . . 3  |-  ( c  =  C  ->  (
( c Full  d ) 
C_  ( c  Func  d )  <->  ( C Full  d
)  C_  ( C  Func  d ) ) )
4 oveq2 6099 . . . 4  |-  ( d  =  D  ->  ( C Full  d )  =  ( C Full  D ) )
5 oveq2 6099 . . . 4  |-  ( d  =  D  ->  ( C  Func  d )  =  ( C  Func  D
) )
64, 5sseq12d 3385 . . 3  |-  ( d  =  D  ->  (
( C Full  d )  C_  ( C  Func  d
)  <->  ( C Full  D
)  C_  ( C  Func  D ) ) )
7 ovex 6116 . . . . . 6  |-  ( c 
Func  d )  e. 
_V
8 simpl 457 . . . . . . . 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 4616 . . . . . . 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 4353 . . . . . . 7  |-  { <. f ,  g >.  |  f ( c  Func  d
) g }  C_  ( c  Func  d
)
119, 10sstri 3365 . . . . . 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 4437 . . . . 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 14814 . . . . . 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 6213 . . . . 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 1303 . . . 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 3406 . . 3  |-  ( ( c  e.  Cat  /\  d  e.  Cat )  ->  ( c Full  d ) 
C_  ( c  Func  d ) )
173, 6, 16vtocl2ga 3038 . 2  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Full  D ) 
C_  ( C  Func  D ) )
1813mpt2ndm0 6739 . . 3  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Full  D )  =  (/) )
19 0ss 3666 . . 3  |-  (/)  C_  ( C  Func  D )
2018, 19syl6eqss 3406 . 2  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Full  D ) 
C_  ( C  Func  D ) )
2117, 20pm2.61i 164 1  |-  ( C Full 
D )  C_  ( C  Func  D )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972    C_ wss 3328   (/)c0 3637   class class class wbr 4292   {copab 4349   ran crn 4841   ` cfv 5418  (class class class)co 6091   Basecbs 14174   Hom chom 14249   Catccat 14602    Func cfunc 14764   Full cful 14812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-full 14814
This theorem is referenced by:  relfull  14818  isfull  14820  fulloppc  14832  cofull  14844  catcisolem  14974  catciso  14975
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