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 ) ) |