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 |
|
caofid1.4 |
|- ( ph -> C e. X ) |
5 |
|
caofid2.5 |
|- ( ( ph /\ x e. S ) -> ( B R x ) = C ) |
6 |
|
fnconstg |
|- ( B e. W -> ( A X. { B } ) Fn A ) |
7 |
3 6
|
syl |
|- ( ph -> ( A X. { B } ) Fn A ) |
8 |
2
|
ffnd |
|- ( ph -> F Fn A ) |
9 |
|
fnconstg |
|- ( C e. X -> ( A X. { C } ) Fn A ) |
10 |
4 9
|
syl |
|- ( ph -> ( A X. { C } ) Fn A ) |
11 |
|
fvconst2g |
|- ( ( B e. W /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B ) |
12 |
3 11
|
sylan |
|- ( ( ph /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B ) |
13 |
|
eqidd |
|- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) ) |
14 |
5
|
ralrimiva |
|- ( ph -> A. x e. S ( B R x ) = C ) |
15 |
2
|
ffvelrnda |
|- ( ( ph /\ w e. A ) -> ( F ` w ) e. S ) |
16 |
|
oveq2 |
|- ( x = ( F ` w ) -> ( B R x ) = ( B R ( F ` w ) ) ) |
17 |
16
|
eqeq1d |
|- ( x = ( F ` w ) -> ( ( B R x ) = C <-> ( B R ( F ` w ) ) = C ) ) |
18 |
17
|
rspccva |
|- ( ( A. x e. S ( B R x ) = C /\ ( F ` w ) e. S ) -> ( B R ( F ` w ) ) = C ) |
19 |
14 15 18
|
syl2an2r |
|- ( ( ph /\ w e. A ) -> ( B R ( F ` w ) ) = C ) |
20 |
|
fvconst2g |
|- ( ( C e. X /\ w e. A ) -> ( ( A X. { C } ) ` w ) = C ) |
21 |
4 20
|
sylan |
|- ( ( ph /\ w e. A ) -> ( ( A X. { C } ) ` w ) = C ) |
22 |
19 21
|
eqtr4d |
|- ( ( ph /\ w e. A ) -> ( B R ( F ` w ) ) = ( ( A X. { C } ) ` w ) ) |
23 |
1 7 8 10 12 13 22
|
offveq |
|- ( ph -> ( ( A X. { B } ) oF R F ) = ( A X. { C } ) ) |