Step |
Hyp |
Ref |
Expression |
1 |
|
prdsbasmpt2.y |
|- Y = ( S Xs_ ( x e. I |-> R ) ) |
2 |
|
prdsbasmpt2.b |
|- B = ( Base ` Y ) |
3 |
|
prdsbasmpt2.s |
|- ( ph -> S e. V ) |
4 |
|
prdsbasmpt2.i |
|- ( ph -> I e. W ) |
5 |
|
prdsbasmpt2.r |
|- ( ph -> A. x e. I R e. X ) |
6 |
|
prdsbasmpt2.k |
|- K = ( Base ` R ) |
7 |
|
prdsbascl.f |
|- ( ph -> F e. B ) |
8 |
|
eqid |
|- ( x e. I |-> R ) = ( x e. I |-> R ) |
9 |
8
|
fnmpt |
|- ( A. x e. I R e. X -> ( x e. I |-> R ) Fn I ) |
10 |
5 9
|
syl |
|- ( ph -> ( x e. I |-> R ) Fn I ) |
11 |
1 2 3 4 10 7
|
prdsbasfn |
|- ( ph -> F Fn I ) |
12 |
|
dffn5 |
|- ( F Fn I <-> F = ( x e. I |-> ( F ` x ) ) ) |
13 |
11 12
|
sylib |
|- ( ph -> F = ( x e. I |-> ( F ` x ) ) ) |
14 |
13 7
|
eqeltrrd |
|- ( ph -> ( x e. I |-> ( F ` x ) ) e. B ) |
15 |
1 2 3 4 5 6
|
prdsbasmpt2 |
|- ( ph -> ( ( x e. I |-> ( F ` x ) ) e. B <-> A. x e. I ( F ` x ) e. K ) ) |
16 |
14 15
|
mpbid |
|- ( ph -> A. x e. I ( F ` x ) e. K ) |