| Step |
Hyp |
Ref |
Expression |
| 1 |
|
curf2.g |
|- G = ( <. C , D >. curryF F ) |
| 2 |
|
curf2.a |
|- A = ( Base ` C ) |
| 3 |
|
curf2.c |
|- ( ph -> C e. Cat ) |
| 4 |
|
curf2.d |
|- ( ph -> D e. Cat ) |
| 5 |
|
curf2.f |
|- ( ph -> F e. ( ( C Xc. D ) Func E ) ) |
| 6 |
|
curf2.b |
|- B = ( Base ` D ) |
| 7 |
|
curf2.h |
|- H = ( Hom ` C ) |
| 8 |
|
curf2.i |
|- I = ( Id ` D ) |
| 9 |
|
curf2.x |
|- ( ph -> X e. A ) |
| 10 |
|
curf2.y |
|- ( ph -> Y e. A ) |
| 11 |
|
curf2.k |
|- ( ph -> K e. ( X H Y ) ) |
| 12 |
|
curf2.l |
|- L = ( ( X ( 2nd ` G ) Y ) ` K ) |
| 13 |
|
curf2.z |
|- ( ph -> Z e. B ) |
| 14 |
1 2 3 4 5 6 7 8 9 10 11 12
|
curf2 |
|- ( ph -> L = ( z e. B |-> ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) ) ) |
| 15 |
|
simpr |
|- ( ( ph /\ z = Z ) -> z = Z ) |
| 16 |
15
|
opeq2d |
|- ( ( ph /\ z = Z ) -> <. X , z >. = <. X , Z >. ) |
| 17 |
15
|
opeq2d |
|- ( ( ph /\ z = Z ) -> <. Y , z >. = <. Y , Z >. ) |
| 18 |
16 17
|
oveq12d |
|- ( ( ph /\ z = Z ) -> ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) = ( <. X , Z >. ( 2nd ` F ) <. Y , Z >. ) ) |
| 19 |
|
eqidd |
|- ( ( ph /\ z = Z ) -> K = K ) |
| 20 |
15
|
fveq2d |
|- ( ( ph /\ z = Z ) -> ( I ` z ) = ( I ` Z ) ) |
| 21 |
18 19 20
|
oveq123d |
|- ( ( ph /\ z = Z ) -> ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) = ( K ( <. X , Z >. ( 2nd ` F ) <. Y , Z >. ) ( I ` Z ) ) ) |
| 22 |
|
ovexd |
|- ( ph -> ( K ( <. X , Z >. ( 2nd ` F ) <. Y , Z >. ) ( I ` Z ) ) e. _V ) |
| 23 |
14 21 13 22
|
fvmptd |
|- ( ph -> ( L ` Z ) = ( K ( <. X , Z >. ( 2nd ` F ) <. Y , Z >. ) ( I ` Z ) ) ) |