Step |
Hyp |
Ref |
Expression |
1 |
|
df-inito |
|- InitO = ( c e. Cat |-> { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) } ) |
2 |
|
eqid |
|- ( oppCat ` c ) = ( oppCat ` c ) |
3 |
2
|
oppccat |
|- ( c e. Cat -> ( oppCat ` c ) e. Cat ) |
4 |
|
eqid |
|- ( Base ` c ) = ( Base ` c ) |
5 |
2 4
|
oppcbas |
|- ( Base ` c ) = ( Base ` ( oppCat ` c ) ) |
6 |
|
eqid |
|- ( Hom ` ( oppCat ` c ) ) = ( Hom ` ( oppCat ` c ) ) |
7 |
3 5 6
|
termoval |
|- ( c e. Cat -> ( TermO ` ( oppCat ` c ) ) = { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( b ( Hom ` ( oppCat ` c ) ) a ) } ) |
8 |
|
eqid |
|- ( Hom ` c ) = ( Hom ` c ) |
9 |
8 2
|
oppchom |
|- ( b ( Hom ` ( oppCat ` c ) ) a ) = ( a ( Hom ` c ) b ) |
10 |
9
|
eleq2i |
|- ( h e. ( b ( Hom ` ( oppCat ` c ) ) a ) <-> h e. ( a ( Hom ` c ) b ) ) |
11 |
10
|
eubii |
|- ( E! h h e. ( b ( Hom ` ( oppCat ` c ) ) a ) <-> E! h h e. ( a ( Hom ` c ) b ) ) |
12 |
11
|
ralbii |
|- ( A. b e. ( Base ` c ) E! h h e. ( b ( Hom ` ( oppCat ` c ) ) a ) <-> A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) ) |
13 |
12
|
rabbii |
|- { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( b ( Hom ` ( oppCat ` c ) ) a ) } = { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) } |
14 |
7 13
|
eqtrdi |
|- ( c e. Cat -> ( TermO ` ( oppCat ` c ) ) = { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) } ) |
15 |
14
|
mpteq2ia |
|- ( c e. Cat |-> ( TermO ` ( oppCat ` c ) ) ) = ( c e. Cat |-> { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) } ) |
16 |
1 15
|
eqtr4i |
|- InitO = ( c e. Cat |-> ( TermO ` ( oppCat ` c ) ) ) |