| Step |
Hyp |
Ref |
Expression |
| 1 |
|
curfval.g |
|- G = ( <. C , D >. curryF F ) |
| 2 |
|
curfval.a |
|- A = ( Base ` C ) |
| 3 |
|
curfval.c |
|- ( ph -> C e. Cat ) |
| 4 |
|
curfval.d |
|- ( ph -> D e. Cat ) |
| 5 |
|
curfval.f |
|- ( ph -> F e. ( ( C Xc. D ) Func E ) ) |
| 6 |
|
curfval.b |
|- B = ( Base ` D ) |
| 7 |
|
curf1.x |
|- ( ph -> X e. A ) |
| 8 |
|
curf1.k |
|- K = ( ( 1st ` G ) ` X ) |
| 9 |
|
curf11.y |
|- ( ph -> Y e. B ) |
| 10 |
|
eqid |
|- ( Hom ` D ) = ( Hom ` D ) |
| 11 |
|
eqid |
|- ( Id ` C ) = ( Id ` C ) |
| 12 |
1 2 3 4 5 6 7 8 10 11
|
curf1 |
|- ( ph -> K = <. ( y e. B |-> ( X ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) >. ) |
| 13 |
6
|
fvexi |
|- B e. _V |
| 14 |
13
|
mptex |
|- ( y e. B |-> ( X ( 1st ` F ) y ) ) e. _V |
| 15 |
13 13
|
mpoex |
|- ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) e. _V |
| 16 |
14 15
|
op1std |
|- ( K = <. ( y e. B |-> ( X ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` X ) ( <. X , y >. ( 2nd ` F ) <. X , z >. ) g ) ) ) >. -> ( 1st ` K ) = ( y e. B |-> ( X ( 1st ` F ) y ) ) ) |
| 17 |
12 16
|
syl |
|- ( ph -> ( 1st ` K ) = ( y e. B |-> ( X ( 1st ` F ) y ) ) ) |
| 18 |
|
simpr |
|- ( ( ph /\ y = Y ) -> y = Y ) |
| 19 |
18
|
oveq2d |
|- ( ( ph /\ y = Y ) -> ( X ( 1st ` F ) y ) = ( X ( 1st ` F ) Y ) ) |
| 20 |
|
ovexd |
|- ( ph -> ( X ( 1st ` F ) Y ) e. _V ) |
| 21 |
17 19 9 20
|
fvmptd |
|- ( ph -> ( ( 1st ` K ) ` Y ) = ( X ( 1st ` F ) Y ) ) |