Step |
Hyp |
Ref |
Expression |
1 |
|
homarcl.h |
|- H = ( HomA ` C ) |
2 |
|
homafval.b |
|- B = ( Base ` C ) |
3 |
|
homafval.c |
|- ( ph -> C e. Cat ) |
4 |
|
homafval.j |
|- J = ( Hom ` C ) |
5 |
|
fveq2 |
|- ( c = C -> ( Base ` c ) = ( Base ` C ) ) |
6 |
5 2
|
eqtr4di |
|- ( c = C -> ( Base ` c ) = B ) |
7 |
6
|
sqxpeqd |
|- ( c = C -> ( ( Base ` c ) X. ( Base ` c ) ) = ( B X. B ) ) |
8 |
|
fveq2 |
|- ( c = C -> ( Hom ` c ) = ( Hom ` C ) ) |
9 |
8 4
|
eqtr4di |
|- ( c = C -> ( Hom ` c ) = J ) |
10 |
9
|
fveq1d |
|- ( c = C -> ( ( Hom ` c ) ` x ) = ( J ` x ) ) |
11 |
10
|
xpeq2d |
|- ( c = C -> ( { x } X. ( ( Hom ` c ) ` x ) ) = ( { x } X. ( J ` x ) ) ) |
12 |
7 11
|
mpteq12dv |
|- ( c = C -> ( x e. ( ( Base ` c ) X. ( Base ` c ) ) |-> ( { x } X. ( ( Hom ` c ) ` x ) ) ) = ( x e. ( B X. B ) |-> ( { x } X. ( J ` x ) ) ) ) |
13 |
|
df-homa |
|- HomA = ( c e. Cat |-> ( x e. ( ( Base ` c ) X. ( Base ` c ) ) |-> ( { x } X. ( ( Hom ` c ) ` x ) ) ) ) |
14 |
2
|
fvexi |
|- B e. _V |
15 |
14 14
|
xpex |
|- ( B X. B ) e. _V |
16 |
15
|
mptex |
|- ( x e. ( B X. B ) |-> ( { x } X. ( J ` x ) ) ) e. _V |
17 |
12 13 16
|
fvmpt |
|- ( C e. Cat -> ( HomA ` C ) = ( x e. ( B X. B ) |-> ( { x } X. ( J ` x ) ) ) ) |
18 |
3 17
|
syl |
|- ( ph -> ( HomA ` C ) = ( x e. ( B X. B ) |-> ( { x } X. ( J ` x ) ) ) ) |
19 |
1 18
|
eqtrid |
|- ( ph -> H = ( x e. ( B X. B ) |-> ( { x } X. ( J ` x ) ) ) ) |