Step |
Hyp |
Ref |
Expression |
1 |
|
caofref.1 |
|- ( ph -> A e. V ) |
2 |
|
caofref.2 |
|- ( ph -> F : A --> S ) |
3 |
|
caofcom.3 |
|- ( ph -> G : A --> S ) |
4 |
|
caofcom.4 |
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x R y ) = ( y R x ) ) |
5 |
2
|
ffvelrnda |
|- ( ( ph /\ w e. A ) -> ( F ` w ) e. S ) |
6 |
3
|
ffvelrnda |
|- ( ( ph /\ w e. A ) -> ( G ` w ) e. S ) |
7 |
5 6
|
jca |
|- ( ( ph /\ w e. A ) -> ( ( F ` w ) e. S /\ ( G ` w ) e. S ) ) |
8 |
4
|
caovcomg |
|- ( ( ph /\ ( ( F ` w ) e. S /\ ( G ` w ) e. S ) ) -> ( ( F ` w ) R ( G ` w ) ) = ( ( G ` w ) R ( F ` w ) ) ) |
9 |
7 8
|
syldan |
|- ( ( ph /\ w e. A ) -> ( ( F ` w ) R ( G ` w ) ) = ( ( G ` w ) R ( F ` w ) ) ) |
10 |
9
|
mpteq2dva |
|- ( ph -> ( w e. A |-> ( ( F ` w ) R ( G ` w ) ) ) = ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) ) |
11 |
2
|
feqmptd |
|- ( ph -> F = ( w e. A |-> ( F ` w ) ) ) |
12 |
3
|
feqmptd |
|- ( ph -> G = ( w e. A |-> ( G ` w ) ) ) |
13 |
1 5 6 11 12
|
offval2 |
|- ( ph -> ( F oF R G ) = ( w e. A |-> ( ( F ` w ) R ( G ` w ) ) ) ) |
14 |
1 6 5 12 11
|
offval2 |
|- ( ph -> ( G oF R F ) = ( w e. A |-> ( ( G ` w ) R ( F ` w ) ) ) ) |
15 |
10 13 14
|
3eqtr4d |
|- ( ph -> ( F oF R G ) = ( G oF R F ) ) |