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 |
|
fvex |
|- ( Base ` c ) e. _V |
8 |
|
fvex |
|- ( Base ` d ) e. _V |
9 |
7 8
|
xpex |
|- ( ( Base ` c ) X. ( Base ` d ) ) e. _V |
10 |
9
|
a1i |
|- ( ( c = C /\ d = D ) -> ( ( Base ` c ) X. ( Base ` d ) ) e. _V ) |
11 |
|
simpl |
|- ( ( c = C /\ d = D ) -> c = C ) |
12 |
11
|
fveq2d |
|- ( ( c = C /\ d = D ) -> ( Base ` c ) = ( Base ` C ) ) |
13 |
|
simpr |
|- ( ( c = C /\ d = D ) -> d = D ) |
14 |
13
|
fveq2d |
|- ( ( c = C /\ d = D ) -> ( Base ` d ) = ( Base ` D ) ) |
15 |
12 14
|
xpeq12d |
|- ( ( c = C /\ d = D ) -> ( ( Base ` c ) X. ( Base ` d ) ) = ( ( Base ` C ) X. ( Base ` D ) ) ) |
16 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
17 |
|
eqid |
|- ( Base ` D ) = ( Base ` D ) |
18 |
1 16 17
|
xpcbas |
|- ( ( Base ` C ) X. ( Base ` D ) ) = ( Base ` T ) |
19 |
18 2
|
eqtr4i |
|- ( ( Base ` C ) X. ( Base ` D ) ) = B |
20 |
15 19
|
eqtrdi |
|- ( ( c = C /\ d = D ) -> ( ( Base ` c ) X. ( Base ` d ) ) = B ) |
21 |
|
simpr |
|- ( ( ( c = C /\ d = D ) /\ b = B ) -> b = B ) |
22 |
21
|
reseq2d |
|- ( ( ( c = C /\ d = D ) /\ b = B ) -> ( 2nd |` b ) = ( 2nd |` B ) ) |
23 |
|
simpll |
|- ( ( ( c = C /\ d = D ) /\ b = B ) -> c = C ) |
24 |
|
simplr |
|- ( ( ( c = C /\ d = D ) /\ b = B ) -> d = D ) |
25 |
23 24
|
oveq12d |
|- ( ( ( c = C /\ d = D ) /\ b = B ) -> ( c Xc. d ) = ( C Xc. D ) ) |
26 |
25 1
|
eqtr4di |
|- ( ( ( c = C /\ d = D ) /\ b = B ) -> ( c Xc. d ) = T ) |
27 |
26
|
fveq2d |
|- ( ( ( c = C /\ d = D ) /\ b = B ) -> ( Hom ` ( c Xc. d ) ) = ( Hom ` T ) ) |
28 |
27 3
|
eqtr4di |
|- ( ( ( c = C /\ d = D ) /\ b = B ) -> ( Hom ` ( c Xc. d ) ) = H ) |
29 |
28
|
oveqd |
|- ( ( ( c = C /\ d = D ) /\ b = B ) -> ( x ( Hom ` ( c Xc. d ) ) y ) = ( x H y ) ) |
30 |
29
|
reseq2d |
|- ( ( ( c = C /\ d = D ) /\ b = B ) -> ( 2nd |` ( x ( Hom ` ( c Xc. d ) ) y ) ) = ( 2nd |` ( x H y ) ) ) |
31 |
21 21 30
|
mpoeq123dv |
|- ( ( ( c = C /\ d = D ) /\ b = B ) -> ( x e. b , y e. b |-> ( 2nd |` ( x ( Hom ` ( c Xc. d ) ) y ) ) ) = ( x e. B , y e. B |-> ( 2nd |` ( x H y ) ) ) ) |
32 |
22 31
|
opeq12d |
|- ( ( ( c = C /\ d = D ) /\ b = B ) -> <. ( 2nd |` b ) , ( x e. b , y e. b |-> ( 2nd |` ( x ( Hom ` ( c Xc. d ) ) y ) ) ) >. = <. ( 2nd |` B ) , ( x e. B , y e. B |-> ( 2nd |` ( x H y ) ) ) >. ) |
33 |
10 20 32
|
csbied2 |
|- ( ( c = C /\ d = D ) -> [_ ( ( Base ` c ) X. ( Base ` d ) ) / b ]_ <. ( 2nd |` b ) , ( x e. b , y e. b |-> ( 2nd |` ( x ( Hom ` ( c Xc. d ) ) y ) ) ) >. = <. ( 2nd |` B ) , ( x e. B , y e. B |-> ( 2nd |` ( x H y ) ) ) >. ) |
34 |
|
df-2ndf |
|- 2ndF = ( c e. Cat , d e. Cat |-> [_ ( ( Base ` c ) X. ( Base ` d ) ) / b ]_ <. ( 2nd |` b ) , ( x e. b , y e. b |-> ( 2nd |` ( x ( Hom ` ( c Xc. d ) ) y ) ) ) >. ) |
35 |
|
opex |
|- <. ( 2nd |` B ) , ( x e. B , y e. B |-> ( 2nd |` ( x H y ) ) ) >. e. _V |
36 |
33 34 35
|
ovmpoa |
|- ( ( C e. Cat /\ D e. Cat ) -> ( C 2ndF D ) = <. ( 2nd |` B ) , ( x e. B , y e. B |-> ( 2nd |` ( x H y ) ) ) >. ) |
37 |
4 5 36
|
syl2anc |
|- ( ph -> ( C 2ndF D ) = <. ( 2nd |` B ) , ( x e. B , y e. B |-> ( 2nd |` ( x H y ) ) ) >. ) |
38 |
6 37
|
eqtrid |
|- ( ph -> Q = <. ( 2nd |` B ) , ( x e. B , y e. B |-> ( 2nd |` ( x H y ) ) ) >. ) |