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Theorem isfull 16570
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 16566 . . 3 |- ( C Full D ) C_ ( C Func D )
21ssbri 4697 . 2 |- ( F ( C Full D ) G -> F ( C Func D ) G )
3 df-br 4654 . . . . . . 7 |- ( F ( C Func D ) G <-> <. F , G >. e. ( C Func D ) )
4 funcrcl 16523 . . . . . . 7 |- ( <. F , G >. e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
53, 4sylbi 207 . . . . . 6 |- ( F ( C Func D ) G -> ( C e. Cat /\ D e. Cat ) )
6 oveq12 6659 . . . . . . . . . 10 |- ( ( c = C /\ d = D ) -> ( c Func d ) = ( C Func D ) )
76breqd 4664 . . . . . . . . 9 |- ( ( c = C /\ d = D ) -> ( f ( c Func d ) g <-> f ( C Func D ) g ) )
8 simpl 473 . . . . . . . . . . . 12 |- ( ( c = C /\ d = D ) -> c = C )
98fveq2d 6195 . . . . . . . . . . 11 |- ( ( c = C /\ d = D ) -> ( Base ` c ) = ( Base ` C ) )
10 isfull.b . . . . . . . . . . 11 |- B = ( Base ` C )
119, 10syl6eqr 2674 . . . . . . . . . 10 |- ( ( c = C /\ d = D ) -> ( Base ` c ) = B )
12 simpr 477 . . . . . . . . . . . . . . 15 |- ( ( c = C /\ d = D ) -> d = D )
1312fveq2d 6195 . . . . . . . . . . . . . 14 |- ( ( c = C /\ d = D ) -> ( Hom ` d ) = ( Hom ` D ) )
14 isfull.j . . . . . . . . . . . . . 14 |- J = ( Hom ` D )
1513, 14syl6eqr 2674 . . . . . . . . . . . . 13 |- ( ( c = C /\ d = D ) -> ( Hom ` d ) = J )
1615oveqd 6667 . . . . . . . . . . . 12 |- ( ( c = C /\ d = D ) -> ( ( f ` x ) ( Hom ` d ) ( f ` y ) ) = ( ( f ` x ) J ( f ` y ) ) )
1716eqeq2d 2632 . . . . . . . . . . 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 3152 . . . . . . . . . 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 3152 . . . . . . . . 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 747 . . . . . . . 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 4716 . . . . . . 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 16564 . . . . . . 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 6678 . . . . . . . 8 |- ( C Func D ) e. _V
24 simpl 473 . . . . . . . . . 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 5003 . . . . . . . . 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 4714 . . . . . . . . 9 |- { <. f , g >. | f ( C Func D ) g } C_ ( C Func D )
2725, 26sstri 3612 . . . . . . . 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 4803 . . . . . . 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 6791 . . . . . 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 17 . . . . 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 4664 . . . 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 16522 . . . . . 6 |- Rel ( C Func D )
33 brrelex12 5155 . . . . . 6 |- ( ( Rel ( C Func D ) /\ F ( C Func D ) G ) -> ( F e. _V /\ G e. _V ) )
3432, 33mpan 706 . . . . 5 |- ( F ( C Func D ) G -> ( F e. _V /\ G e. _V ) )
35 breq12 4658 . . . . . . 7 |- ( ( f = F /\ g = G ) -> ( f ( C Func D ) g <-> F ( C Func D ) G ) )
36 simpr 477 . . . . . . . . . . 11 |- ( ( f = F /\ g = G ) -> g = G )
3736oveqd 6667 . . . . . . . . . 10 |- ( ( f = F /\ g = G ) -> ( x g y ) = ( x G y ) )
3837rneqd 5353 . . . . . . . . 9 |- ( ( f = F /\ g = G ) -> ran ( x g y ) = ran ( x G y ) )
39 simpl 473 . . . . . . . . . . 11 |- ( ( f = F /\ g = G ) -> f = F )
4039fveq1d 6193 . . . . . . . . . 10 |- ( ( f = F /\ g = G ) -> ( f ` x ) = ( F ` x ) )
4139fveq1d 6193 . . . . . . . . . 10 |- ( ( f = F /\ g = G ) -> ( f ` y ) = ( F ` y ) )
4240, 41oveq12d 6668 . . . . . . . . 9 |- ( ( f = F /\ g = G ) -> ( ( f ` x ) J ( f ` y ) ) = ( ( F ` x ) J ( F ` y ) ) )
4338, 42eqeq12d 2637 . . . . . . . 8 |- ( ( f = F /\ g = G ) -> ( ran ( x g y ) = ( ( f ` x ) J ( f ` y ) ) <-> ran ( x G y ) = ( ( F ` x ) J ( F ` y ) ) ) )
44432ralbidv 2989 . . . . . . 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 747 . . . . . 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 2622 . . . . . 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 4989 . . . . 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 17 . . . 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 268 . . 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 924 . 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 674 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 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.cop 4183   class class class wbr 4653   {copab 4712   ran crn 5115   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952   Catccat 16325   Func cfunc 16514   Full cful 16562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-func 16518  df-full 16564
This theorem is referenced by:  isfull2  16571  fullpropd  16580  fulloppc  16582  fullres2c  16599
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