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Theorem fullfunc 14093
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 6080 . . . 4  |-  ( c  =  C  ->  (
c Full  d )  =  ( C Full  d ) )
2 oveq1 6080 . . . 4  |-  ( c  =  C  ->  (
c  Func  d )  =  ( C  Func  d ) )
31, 2sseq12d 3369 . . 3  |-  ( c  =  C  ->  (
( c Full  d ) 
C_  ( c  Func  d )  <->  ( C Full  d
)  C_  ( C  Func  d ) ) )
4 oveq2 6081 . . . 4  |-  ( d  =  D  ->  ( C Full  d )  =  ( C Full  D ) )
5 oveq2 6081 . . . 4  |-  ( d  =  D  ->  ( C  Func  d )  =  ( C  Func  D
) )
64, 5sseq12d 3369 . . 3  |-  ( d  =  D  ->  (
( C Full  d )  C_  ( C  Func  d
)  <->  ( C Full  D
)  C_  ( C  Func  D ) ) )
7 ovex 6098 . . . . . 6  |-  ( c 
Func  d )  e. 
_V
8 simpl 444 . . . . . . . 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 4474 . . . . . . 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 4261 . . . . . . 7  |-  { <. f ,  g >.  |  f ( c  Func  d
) g }  C_  ( c  Func  d
)
119, 10sstri 3349 . . . . . 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 4340 . . . . 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 14091 . . . . . 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 6188 . . . . 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 1268 . . . 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 3390 . . 3  |-  ( ( c  e.  Cat  /\  d  e.  Cat )  ->  ( c Full  d ) 
C_  ( c  Func  d ) )
173, 6, 16vtocl2ga 3011 . 2  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Full  D ) 
C_  ( C  Func  D ) )
1813mpt2ndm0 6465 . . 3  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Full  D )  =  (/) )
19 0ss 3648 . . 3  |-  (/)  C_  ( C  Func  D )
2018, 19syl6eqss 3390 . 2  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Full  D ) 
C_  ( C  Func  D ) )
2117, 20pm2.61i 158 1  |-  ( C Full 
D )  C_  ( C  Func  D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    C_ wss 3312   (/)c0 3620   class class class wbr 4204   {copab 4257   ran crn 4871   ` cfv 5446  (class class class)co 6073   Basecbs 13459    Hom chom 13530   Catccat 13879    Func cfunc 14041   Full cful 14089
This theorem is referenced by:  relfull  14095  isfull  14097  fulloppc  14109  cofull  14121  catcisolem  14251  catciso  14252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-full 14091
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