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 |
|
2ndfval.p |
|- Q = ( C 2ndF D ) |
7 |
|
2ndf1.p |
|- ( ph -> R e. B ) |
8 |
|
2ndf2.p |
|- ( ph -> S e. B ) |
9 |
1 2 3 4 5 6
|
2ndfval |
|- ( ph -> Q = <. ( 2nd |` B ) , ( x e. B , y e. B |-> ( 2nd |` ( x H y ) ) ) >. ) |
10 |
|
fo2nd |
|- 2nd : _V -onto-> _V |
11 |
|
fofun |
|- ( 2nd : _V -onto-> _V -> Fun 2nd ) |
12 |
10 11
|
ax-mp |
|- Fun 2nd |
13 |
2
|
fvexi |
|- B e. _V |
14 |
|
resfunexg |
|- ( ( Fun 2nd /\ B e. _V ) -> ( 2nd |` B ) e. _V ) |
15 |
12 13 14
|
mp2an |
|- ( 2nd |` B ) e. _V |
16 |
13 13
|
mpoex |
|- ( x e. B , y e. B |-> ( 2nd |` ( x H y ) ) ) e. _V |
17 |
15 16
|
op2ndd |
|- ( Q = <. ( 2nd |` B ) , ( x e. B , y e. B |-> ( 2nd |` ( x H y ) ) ) >. -> ( 2nd ` Q ) = ( x e. B , y e. B |-> ( 2nd |` ( x H y ) ) ) ) |
18 |
9 17
|
syl |
|- ( ph -> ( 2nd ` Q ) = ( x e. B , y e. B |-> ( 2nd |` ( 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 ) ) -> ( 2nd |` ( x H y ) ) = ( 2nd |` ( R H S ) ) ) |
23 |
|
ovex |
|- ( R H S ) e. _V |
24 |
|
resfunexg |
|- ( ( Fun 2nd /\ ( R H S ) e. _V ) -> ( 2nd |` ( R H S ) ) e. _V ) |
25 |
12 23 24
|
mp2an |
|- ( 2nd |` ( R H S ) ) e. _V |
26 |
25
|
a1i |
|- ( ph -> ( 2nd |` ( R H S ) ) e. _V ) |
27 |
18 22 7 8 26
|
ovmpod |
|- ( ph -> ( R ( 2nd ` Q ) S ) = ( 2nd |` ( R H S ) ) ) |