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Theorem isfull 15326
Description: Value of the set of full functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
isfull.b  |-  B  =  ( Base `  C
)
isfull.j  |-  J  =  ( Hom  `  D
)
Assertion
Ref Expression
isfull  |-  ( F ( C Full  D ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  B  A. y  e.  B  ran  ( x G y )  =  ( ( F `  x ) J ( F `  y ) ) ) )
Distinct variable groups:    x, y, B    x, C, y    x, D, y    x, J, y   
x, F, y    x, G, y

Proof of Theorem isfull
Dummy variables  c 
d  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fullfunc 15322 . . 3  |-  ( C Full 
D )  C_  ( C  Func  D )
21ssbri 4498 . 2  |-  ( F ( C Full  D ) G  ->  F ( C  Func  D ) G )
3 df-br 4457 . . . . . . 7  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
4 funcrcl 15279 . . . . . . 7  |-  ( <. F ,  G >.  e.  ( C  Func  D
)  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
53, 4sylbi 195 . . . . . 6  |-  ( F ( C  Func  D
) G  ->  ( C  e.  Cat  /\  D  e.  Cat ) )
6 oveq12 6305 . . . . . . . . . 10  |-  ( ( c  =  C  /\  d  =  D )  ->  ( c  Func  d
)  =  ( C 
Func  D ) )
76breqd 4467 . . . . . . . . 9  |-  ( ( c  =  C  /\  d  =  D )  ->  ( f ( c 
Func  d ) g  <-> 
f ( C  Func  D ) g ) )
8 simpl 457 . . . . . . . . . . . 12  |-  ( ( c  =  C  /\  d  =  D )  ->  c  =  C )
98fveq2d 5876 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  d  =  D )  ->  ( Base `  c
)  =  ( Base `  C ) )
10 isfull.b . . . . . . . . . . 11  |-  B  =  ( Base `  C
)
119, 10syl6eqr 2516 . . . . . . . . . 10  |-  ( ( c  =  C  /\  d  =  D )  ->  ( Base `  c
)  =  B )
12 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( c  =  C  /\  d  =  D )  ->  d  =  D )
1312fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( ( c  =  C  /\  d  =  D )  ->  ( Hom  `  d
)  =  ( Hom  `  D ) )
14 isfull.j . . . . . . . . . . . . . 14  |-  J  =  ( Hom  `  D
)
1513, 14syl6eqr 2516 . . . . . . . . . . . . 13  |-  ( ( c  =  C  /\  d  =  D )  ->  ( Hom  `  d
)  =  J )
1615oveqd 6313 . . . . . . . . . . . 12  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ( f `  x ) ( Hom  `  d ) ( f `
 y ) )  =  ( ( f `
 x ) J ( f `  y
) ) )
1716eqeq2d 2471 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ran  ( x g y )  =  ( ( f `  x ) ( Hom  `  d ) ( f `
 y ) )  <->  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) )
1811, 17raleqbidv 3068 . . . . . . . . . 10  |-  ( ( c  =  C  /\  d  =  D )  ->  ( A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) ( Hom  `  d ) ( f `
 y ) )  <->  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) )
1911, 18raleqbidv 3068 . . . . . . . . 9  |-  ( ( c  =  C  /\  d  =  D )  ->  ( A. x  e.  ( Base `  c
) A. y  e.  ( Base `  c
) ran  ( x
g y )  =  ( ( f `  x ) ( Hom  `  d ) ( f `
 y ) )  <->  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) )
207, 19anbi12d 710 . . . . . . . 8  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ( 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  /\  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) ) )
2120opabbidv 4520 . . . . . . 7  |-  ( ( c  =  C  /\  d  =  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 ) ) ) }  =  { <. f ,  g >.  |  ( f ( C  Func  D )
g  /\  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) } )
22 df-full 15320 . . . . . . 7  |- 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 ) ) ) } )
23 ovex 6324 . . . . . . . 8  |-  ( C 
Func  D )  e.  _V
24 simpl 457 . . . . . . . . . 10  |-  ( ( f ( C  Func  D ) g  /\  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) )  ->  f ( C 
Func  D ) g )
2524ssopab2i 4784 . . . . . . . . 9  |-  { <. f ,  g >.  |  ( f ( C  Func  D ) g  /\  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) }  C_  { <. f ,  g >.  |  f ( C  Func  D
) g }
26 opabss 4518 . . . . . . . . 9  |-  { <. f ,  g >.  |  f ( C  Func  D
) g }  C_  ( C  Func  D )
2725, 26sstri 3508 . . . . . . . 8  |-  { <. f ,  g >.  |  ( f ( C  Func  D ) g  /\  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) }  C_  ( C  Func  D )
2823, 27ssexi 4601 . . . . . . 7  |-  { <. f ,  g >.  |  ( f ( C  Func  D ) g  /\  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) }  e.  _V
2921, 22, 28ovmpt2a 6432 . . . . . 6  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Full  D )  =  { <. f ,  g >.  |  ( f ( C  Func  D ) g  /\  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) } )
305, 29syl 16 . . . . 5  |-  ( F ( C  Func  D
) G  ->  ( C Full  D )  =  { <. f ,  g >.  |  ( f ( C  Func  D )
g  /\  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) } )
3130breqd 4467 . . . 4  |-  ( F ( C  Func  D
) G  ->  ( F ( C Full  D
) G  <->  F { <. f ,  g >.  |  ( f ( C  Func  D )
g  /\  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) } G
) )
32 relfunc 15278 . . . . . 6  |-  Rel  ( C  Func  D )
33 brrelex12 5046 . . . . . 6  |-  ( ( Rel  ( C  Func  D )  /\  F ( C  Func  D ) G )  ->  ( F  e.  _V  /\  G  e.  _V ) )
3432, 33mpan 670 . . . . 5  |-  ( F ( C  Func  D
) G  ->  ( F  e.  _V  /\  G  e.  _V ) )
35 breq12 4461 . . . . . . 7  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f ( C 
Func  D ) g  <->  F ( C  Func  D ) G ) )
36 simpr 461 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  g  =  G )  ->  g  =  G )
3736oveqd 6313 . . . . . . . . . 10  |-  ( ( f  =  F  /\  g  =  G )  ->  ( x g y )  =  ( x G y ) )
3837rneqd 5240 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ran  ( x g y )  =  ran  ( x G y ) )
39 simpl 457 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  g  =  G )  ->  f  =  F )
4039fveq1d 5874 . . . . . . . . . 10  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f `  x
)  =  ( F `
 x ) )
4139fveq1d 5874 . . . . . . . . . 10  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f `  y
)  =  ( F `
 y ) )
4240, 41oveq12d 6314 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f `  x ) J ( f `  y ) )  =  ( ( F `  x ) J ( F `  y ) ) )
4338, 42eqeq12d 2479 . . . . . . . 8  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) )  <->  ran  ( x G y )  =  ( ( F `  x
) J ( F `
 y ) ) ) )
44432ralbidv 2901 . . . . . . 7  |-  ( ( f  =  F  /\  g  =  G )  ->  ( A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) )  <->  A. x  e.  B  A. y  e.  B  ran  ( x G y )  =  ( ( F `  x ) J ( F `  y ) ) ) )
4535, 44anbi12d 710 . . . . . 6  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f ( C  Func  D )
g  /\  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) )  <->  ( F
( C  Func  D
) G  /\  A. x  e.  B  A. y  e.  B  ran  ( x G y )  =  ( ( F `  x ) J ( F `  y ) ) ) ) )
46 eqid 2457 . . . . . 6  |-  { <. f ,  g >.  |  ( f ( C  Func  D ) g  /\  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) }  =  { <. f ,  g >.  |  ( f ( C  Func  D ) g  /\  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) }
4745, 46brabga 4770 . . . . 5  |-  ( ( F  e.  _V  /\  G  e.  _V )  ->  ( F { <. f ,  g >.  |  ( f ( C  Func  D ) g  /\  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) } G  <->  ( F
( C  Func  D
) G  /\  A. x  e.  B  A. y  e.  B  ran  ( x G y )  =  ( ( F `  x ) J ( F `  y ) ) ) ) )
4834, 47syl 16 . . . 4  |-  ( F ( C  Func  D
) G  ->  ( F { <. f ,  g
>.  |  ( f
( C  Func  D
) g  /\  A. x  e.  B  A. y  e.  B  ran  ( x g y )  =  ( ( f `  x ) J ( f `  y ) ) ) } G  <->  ( F
( C  Func  D
) G  /\  A. x  e.  B  A. y  e.  B  ran  ( x G y )  =  ( ( F `  x ) J ( F `  y ) ) ) ) )
4931, 48bitrd 253 . . 3  |-  ( F ( C  Func  D
) G  ->  ( F ( C Full  D
) G  <->  ( F
( C  Func  D
) G  /\  A. x  e.  B  A. y  e.  B  ran  ( x G y )  =  ( ( F `  x ) J ( F `  y ) ) ) ) )
5049bianabs 880 . 2  |-  ( F ( C  Func  D
) G  ->  ( F ( C Full  D
) G  <->  A. x  e.  B  A. y  e.  B  ran  ( x G y )  =  ( ( F `  x ) J ( F `  y ) ) ) )
512, 50biadan2 642 1  |-  ( F ( C Full  D ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  B  A. y  e.  B  ran  ( x G y )  =  ( ( F `  x ) J ( F `  y ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109   <.cop 4038   class class class wbr 4456   {copab 4514   ran crn 5009   Rel wrel 5013   ` 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  ax-un 6591
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-csb 3431  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-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-func 15274  df-full 15320
This theorem is referenced by:  isfull2  15327  fullpropd  15336  fulloppc  15338  fullres2c  15355
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