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Theorem fullfunc 15322
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 6303 . . . 4  |-  ( c  =  C  ->  (
c Full  d )  =  ( C Full  d ) )
2 oveq1 6303 . . . 4  |-  ( c  =  C  ->  (
c  Func  d )  =  ( C  Func  d ) )
31, 2sseq12d 3528 . . 3  |-  ( c  =  C  ->  (
( c Full  d ) 
C_  ( c  Func  d )  <->  ( C Full  d
)  C_  ( C  Func  d ) ) )
4 oveq2 6304 . . . 4  |-  ( d  =  D  ->  ( C Full  d )  =  ( C Full  D ) )
5 oveq2 6304 . . . 4  |-  ( d  =  D  ->  ( C  Func  d )  =  ( C  Func  D
) )
64, 5sseq12d 3528 . . 3  |-  ( d  =  D  ->  (
( C Full  d )  C_  ( C  Func  d
)  <->  ( C Full  D
)  C_  ( C  Func  D ) ) )
7 ovex 6324 . . . . . 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 4784 . . . . . . 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 4518 . . . . . . 7  |-  { <. f ,  g >.  |  f ( c  Func  d
) g }  C_  ( c  Func  d
)
119, 10sstri 3508 . . . . . 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 4601 . . . . 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 15320 . . . . . 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 6424 . . . . 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 1313 . . . 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 3549 . . 3  |-  ( ( c  e.  Cat  /\  d  e.  Cat )  ->  ( c Full  d ) 
C_  ( c  Func  d ) )
173, 6, 16vtocl2ga 3175 . 2  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Full  D ) 
C_  ( C  Func  D ) )
1813mpt2ndm0 6515 . . 3  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Full  D )  =  (/) )
19 0ss 3823 . . 3  |-  (/)  C_  ( C  Func  D )
2018, 19syl6eqss 3549 . 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 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109    C_ wss 3471   (/)c0 3793   class class class wbr 4456   {copab 4514   ran crn 5009   ` cfv 5594  (class class class)co 6296   Basecbs 14644   Hom chom 14723   Catccat 15081    Func cfunc 15270   Full cful 15318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-full 15320
This theorem is referenced by:  relfull  15324  isfull  15326  fulloppc  15338  cofull  15350  catcisolem  15512  catciso  15513
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