Step |
Hyp |
Ref |
Expression |
1 |
|
rescbas.d |
|- D = ( C |`cat H ) |
2 |
|
rescbas.b |
|- B = ( Base ` C ) |
3 |
|
rescbas.c |
|- ( ph -> C e. V ) |
4 |
|
rescbas.h |
|- ( ph -> H Fn ( S X. S ) ) |
5 |
|
rescbas.s |
|- ( ph -> S C_ B ) |
6 |
|
baseid |
|- Base = Slot ( Base ` ndx ) |
7 |
|
1re |
|- 1 e. RR |
8 |
|
1nn |
|- 1 e. NN |
9 |
|
4nn0 |
|- 4 e. NN0 |
10 |
|
1nn0 |
|- 1 e. NN0 |
11 |
|
1lt10 |
|- 1 < ; 1 0 |
12 |
8 9 10 11
|
declti |
|- 1 < ; 1 4 |
13 |
7 12
|
ltneii |
|- 1 =/= ; 1 4 |
14 |
|
basendx |
|- ( Base ` ndx ) = 1 |
15 |
|
homndx |
|- ( Hom ` ndx ) = ; 1 4 |
16 |
14 15
|
neeq12i |
|- ( ( Base ` ndx ) =/= ( Hom ` ndx ) <-> 1 =/= ; 1 4 ) |
17 |
13 16
|
mpbir |
|- ( Base ` ndx ) =/= ( Hom ` ndx ) |
18 |
6 17
|
setsnid |
|- ( Base ` ( C |`s S ) ) = ( Base ` ( ( C |`s S ) sSet <. ( Hom ` ndx ) , H >. ) ) |
19 |
|
eqid |
|- ( C |`s S ) = ( C |`s S ) |
20 |
19 2
|
ressbas2 |
|- ( S C_ B -> S = ( Base ` ( C |`s S ) ) ) |
21 |
5 20
|
syl |
|- ( ph -> S = ( Base ` ( C |`s S ) ) ) |
22 |
2
|
fvexi |
|- B e. _V |
23 |
22
|
ssex |
|- ( S C_ B -> S e. _V ) |
24 |
5 23
|
syl |
|- ( ph -> S e. _V ) |
25 |
1 3 24 4
|
rescval2 |
|- ( ph -> D = ( ( C |`s S ) sSet <. ( Hom ` ndx ) , H >. ) ) |
26 |
25
|
fveq2d |
|- ( ph -> ( Base ` D ) = ( Base ` ( ( C |`s S ) sSet <. ( Hom ` ndx ) , H >. ) ) ) |
27 |
18 21 26
|
3eqtr4a |
|- ( ph -> S = ( Base ` D ) ) |