Step |
Hyp |
Ref |
Expression |
1 |
|
fvres |
|- ( c e. Cat -> ( ( oppCat |` Cat ) ` c ) = ( oppCat ` c ) ) |
2 |
1
|
fveq2d |
|- ( c e. Cat -> ( InitO ` ( ( oppCat |` Cat ) ` c ) ) = ( InitO ` ( oppCat ` c ) ) ) |
3 |
2
|
mpteq2ia |
|- ( c e. Cat |-> ( InitO ` ( ( oppCat |` Cat ) ` c ) ) ) = ( c e. Cat |-> ( InitO ` ( oppCat ` c ) ) ) |
4 |
|
initofn |
|- InitO Fn Cat |
5 |
|
dffn2 |
|- ( InitO Fn Cat <-> InitO : Cat --> _V ) |
6 |
4 5
|
mpbi |
|- InitO : Cat --> _V |
7 |
|
oppccatf |
|- ( oppCat |` Cat ) : Cat --> Cat |
8 |
|
fcompt |
|- ( ( InitO : Cat --> _V /\ ( oppCat |` Cat ) : Cat --> Cat ) -> ( InitO o. ( oppCat |` Cat ) ) = ( c e. Cat |-> ( InitO ` ( ( oppCat |` Cat ) ` c ) ) ) ) |
9 |
6 7 8
|
mp2an |
|- ( InitO o. ( oppCat |` Cat ) ) = ( c e. Cat |-> ( InitO ` ( ( oppCat |` Cat ) ` c ) ) ) |
10 |
|
dftermo2 |
|- TermO = ( c e. Cat |-> ( InitO ` ( oppCat ` c ) ) ) |
11 |
3 9 10
|
3eqtr4ri |
|- TermO = ( InitO o. ( oppCat |` Cat ) ) |