Step |
Hyp |
Ref |
Expression |
1 |
|
isfull.b |
|- B = ( Base ` C ) |
2 |
|
isfull.j |
|- J = ( Hom ` D ) |
3 |
|
fullfunc |
|- ( C Full D ) C_ ( C Func D ) |
4 |
3
|
ssbri |
|- ( F ( C Full D ) G -> F ( C Func D ) G ) |
5 |
|
df-br |
|- ( F ( C Func D ) G <-> <. F , G >. e. ( C Func D ) ) |
6 |
|
funcrcl |
|- ( <. F , G >. e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) ) |
7 |
5 6
|
sylbi |
|- ( F ( C Func D ) G -> ( C e. Cat /\ D e. Cat ) ) |
8 |
|
oveq12 |
|- ( ( c = C /\ d = D ) -> ( c Func d ) = ( C Func D ) ) |
9 |
8
|
breqd |
|- ( ( c = C /\ d = D ) -> ( f ( c Func d ) g <-> f ( C Func D ) g ) ) |
10 |
|
simpl |
|- ( ( c = C /\ d = D ) -> c = C ) |
11 |
10
|
fveq2d |
|- ( ( c = C /\ d = D ) -> ( Base ` c ) = ( Base ` C ) ) |
12 |
11 1
|
eqtr4di |
|- ( ( c = C /\ d = D ) -> ( Base ` c ) = B ) |
13 |
|
simpr |
|- ( ( c = C /\ d = D ) -> d = D ) |
14 |
13
|
fveq2d |
|- ( ( c = C /\ d = D ) -> ( Hom ` d ) = ( Hom ` D ) ) |
15 |
14 2
|
eqtr4di |
|- ( ( c = C /\ d = D ) -> ( Hom ` d ) = J ) |
16 |
15
|
oveqd |
|- ( ( c = C /\ d = D ) -> ( ( f ` x ) ( Hom ` d ) ( f ` y ) ) = ( ( f ` x ) J ( f ` y ) ) ) |
17 |
16
|
eqeq2d |
|- ( ( c = C /\ d = D ) -> ( ran ( x g y ) = ( ( f ` x ) ( Hom ` d ) ( f ` y ) ) <-> ran ( x g y ) = ( ( f ` x ) J ( f ` y ) ) ) ) |
18 |
12 17
|
raleqbidv |
|- ( ( 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 ) ) ) ) |
19 |
12 18
|
raleqbidv |
|- ( ( 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 ) ) ) ) |
20 |
9 19
|
anbi12d |
|- ( ( 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 ) ) ) ) ) |
21 |
20
|
opabbidv |
|- ( ( 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 |
|- 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 |
|- ( C Func D ) e. _V |
24 |
|
simpl |
|- ( ( 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 ) |
25 |
24
|
ssopab2i |
|- { <. 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 |
|- { <. f , g >. | f ( C Func D ) g } C_ ( C Func D ) |
27 |
25 26
|
sstri |
|- { <. 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 ) |
28 |
23 27
|
ssexi |
|- { <. 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 |
29 |
21 22 28
|
ovmpoa |
|- ( ( 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 ) ) ) } ) |
30 |
7 29
|
syl |
|- ( 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 ) ) ) } ) |
31 |
30
|
breqd |
|- ( 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 |
|- Rel ( C Func D ) |
33 |
32
|
brrelex12i |
|- ( F ( C Func D ) G -> ( F e. _V /\ G e. _V ) ) |
34 |
|
breq12 |
|- ( ( f = F /\ g = G ) -> ( f ( C Func D ) g <-> F ( C Func D ) G ) ) |
35 |
|
simpr |
|- ( ( f = F /\ g = G ) -> g = G ) |
36 |
35
|
oveqd |
|- ( ( f = F /\ g = G ) -> ( x g y ) = ( x G y ) ) |
37 |
36
|
rneqd |
|- ( ( f = F /\ g = G ) -> ran ( x g y ) = ran ( x G y ) ) |
38 |
|
simpl |
|- ( ( f = F /\ g = G ) -> f = F ) |
39 |
38
|
fveq1d |
|- ( ( f = F /\ g = G ) -> ( f ` x ) = ( F ` x ) ) |
40 |
38
|
fveq1d |
|- ( ( f = F /\ g = G ) -> ( f ` y ) = ( F ` y ) ) |
41 |
39 40
|
oveq12d |
|- ( ( f = F /\ g = G ) -> ( ( f ` x ) J ( f ` y ) ) = ( ( F ` x ) J ( F ` y ) ) ) |
42 |
37 41
|
eqeq12d |
|- ( ( f = F /\ g = G ) -> ( ran ( x g y ) = ( ( f ` x ) J ( f ` y ) ) <-> ran ( x G y ) = ( ( F ` x ) J ( F ` y ) ) ) ) |
43 |
42
|
2ralbidv |
|- ( ( 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 ) ) ) ) |
44 |
34 43
|
anbi12d |
|- ( ( 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 ) ) ) ) ) |
45 |
|
eqid |
|- { <. 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 ) ) ) } |
46 |
44 45
|
brabga |
|- ( ( 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 ) ) ) ) ) |
47 |
33 46
|
syl |
|- ( 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 ) ) ) ) ) |
48 |
31 47
|
bitrd |
|- ( 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 ) ) ) ) ) |
49 |
48
|
bianabs |
|- ( 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 ) ) ) ) |
50 |
4 49
|
biadanii |
|- ( 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 ) ) ) ) |