Step |
Hyp |
Ref |
Expression |
1 |
|
caofref.1 |
|- ( ph -> A e. V ) |
2 |
|
caofref.2 |
|- ( ph -> F : A --> S ) |
3 |
|
caofid0.3 |
|- ( ph -> B e. W ) |
4 |
|
caofid0l.5 |
|- ( ( ph /\ x e. S ) -> ( B R x ) = x ) |
5 |
|
fnconstg |
|- ( B e. W -> ( A X. { B } ) Fn A ) |
6 |
3 5
|
syl |
|- ( ph -> ( A X. { B } ) Fn A ) |
7 |
2
|
ffnd |
|- ( ph -> F Fn A ) |
8 |
|
fvconst2g |
|- ( ( B e. W /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B ) |
9 |
3 8
|
sylan |
|- ( ( ph /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B ) |
10 |
|
eqidd |
|- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) ) |
11 |
4
|
ralrimiva |
|- ( ph -> A. x e. S ( B R x ) = x ) |
12 |
2
|
ffvelrnda |
|- ( ( ph /\ w e. A ) -> ( F ` w ) e. S ) |
13 |
|
oveq2 |
|- ( x = ( F ` w ) -> ( B R x ) = ( B R ( F ` w ) ) ) |
14 |
|
id |
|- ( x = ( F ` w ) -> x = ( F ` w ) ) |
15 |
13 14
|
eqeq12d |
|- ( x = ( F ` w ) -> ( ( B R x ) = x <-> ( B R ( F ` w ) ) = ( F ` w ) ) ) |
16 |
15
|
rspccva |
|- ( ( A. x e. S ( B R x ) = x /\ ( F ` w ) e. S ) -> ( B R ( F ` w ) ) = ( F ` w ) ) |
17 |
11 12 16
|
syl2an2r |
|- ( ( ph /\ w e. A ) -> ( B R ( F ` w ) ) = ( F ` w ) ) |
18 |
1 6 7 7 9 10 17
|
offveq |
|- ( ph -> ( ( A X. { B } ) oF R F ) = F ) |