| Step |
Hyp |
Ref |
Expression |
| 1 |
|
1stfval.t |
|- T = ( C Xc. D ) |
| 2 |
|
1stfval.b |
|- B = ( Base ` T ) |
| 3 |
|
1stfval.h |
|- H = ( Hom ` T ) |
| 4 |
|
1stfval.c |
|- ( ph -> C e. Cat ) |
| 5 |
|
1stfval.d |
|- ( ph -> D e. Cat ) |
| 6 |
|
1stfval.p |
|- P = ( C 1stF D ) |
| 7 |
|
1stf1.p |
|- ( ph -> R e. B ) |
| 8 |
|
1stf2.p |
|- ( ph -> S e. B ) |
| 9 |
1 2 3 4 5 6
|
1stfval |
|- ( ph -> P = <. ( 1st |` B ) , ( x e. B , y e. B |-> ( 1st |` ( x H y ) ) ) >. ) |
| 10 |
|
fo1st |
|- 1st : _V -onto-> _V |
| 11 |
|
fofun |
|- ( 1st : _V -onto-> _V -> Fun 1st ) |
| 12 |
10 11
|
ax-mp |
|- Fun 1st |
| 13 |
2
|
fvexi |
|- B e. _V |
| 14 |
|
resfunexg |
|- ( ( Fun 1st /\ B e. _V ) -> ( 1st |` B ) e. _V ) |
| 15 |
12 13 14
|
mp2an |
|- ( 1st |` B ) e. _V |
| 16 |
13 13
|
mpoex |
|- ( x e. B , y e. B |-> ( 1st |` ( x H y ) ) ) e. _V |
| 17 |
15 16
|
op2ndd |
|- ( P = <. ( 1st |` B ) , ( x e. B , y e. B |-> ( 1st |` ( x H y ) ) ) >. -> ( 2nd ` P ) = ( x e. B , y e. B |-> ( 1st |` ( x H y ) ) ) ) |
| 18 |
9 17
|
syl |
|- ( ph -> ( 2nd ` P ) = ( x e. B , y e. B |-> ( 1st |` ( x H y ) ) ) ) |
| 19 |
|
simprl |
|- ( ( ph /\ ( x = R /\ y = S ) ) -> x = R ) |
| 20 |
|
simprr |
|- ( ( ph /\ ( x = R /\ y = S ) ) -> y = S ) |
| 21 |
19 20
|
oveq12d |
|- ( ( ph /\ ( x = R /\ y = S ) ) -> ( x H y ) = ( R H S ) ) |
| 22 |
21
|
reseq2d |
|- ( ( ph /\ ( x = R /\ y = S ) ) -> ( 1st |` ( x H y ) ) = ( 1st |` ( R H S ) ) ) |
| 23 |
|
ovex |
|- ( R H S ) e. _V |
| 24 |
|
resfunexg |
|- ( ( Fun 1st /\ ( R H S ) e. _V ) -> ( 1st |` ( R H S ) ) e. _V ) |
| 25 |
12 23 24
|
mp2an |
|- ( 1st |` ( R H S ) ) e. _V |
| 26 |
25
|
a1i |
|- ( ph -> ( 1st |` ( R H S ) ) e. _V ) |
| 27 |
18 22 7 8 26
|
ovmpod |
|- ( ph -> ( R ( 2nd ` P ) S ) = ( 1st |` ( R H S ) ) ) |