| Step |
Hyp |
Ref |
Expression |
| 1 |
|
offval.1 |
|- ( ph -> F Fn A ) |
| 2 |
|
offval.2 |
|- ( ph -> G Fn B ) |
| 3 |
|
offval.3 |
|- ( ph -> A e. V ) |
| 4 |
|
offval.4 |
|- ( ph -> B e. W ) |
| 5 |
|
offval.5 |
|- ( A i^i B ) = S |
| 6 |
|
ofval.6 |
|- ( ( ph /\ X e. A ) -> ( F ` X ) = C ) |
| 7 |
|
ofval.7 |
|- ( ( ph /\ X e. B ) -> ( G ` X ) = D ) |
| 8 |
|
eqidd |
|- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) ) |
| 9 |
|
eqidd |
|- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) ) |
| 10 |
1 2 3 4 5 8 9
|
ofrfval |
|- ( ph -> ( F oR R G <-> A. x e. S ( F ` x ) R ( G ` x ) ) ) |
| 11 |
10
|
biimpa |
|- ( ( ph /\ F oR R G ) -> A. x e. S ( F ` x ) R ( G ` x ) ) |
| 12 |
|
fveq2 |
|- ( x = X -> ( F ` x ) = ( F ` X ) ) |
| 13 |
|
fveq2 |
|- ( x = X -> ( G ` x ) = ( G ` X ) ) |
| 14 |
12 13
|
breq12d |
|- ( x = X -> ( ( F ` x ) R ( G ` x ) <-> ( F ` X ) R ( G ` X ) ) ) |
| 15 |
14
|
rspccv |
|- ( A. x e. S ( F ` x ) R ( G ` x ) -> ( X e. S -> ( F ` X ) R ( G ` X ) ) ) |
| 16 |
11 15
|
syl |
|- ( ( ph /\ F oR R G ) -> ( X e. S -> ( F ` X ) R ( G ` X ) ) ) |
| 17 |
16
|
3impia |
|- ( ( ph /\ F oR R G /\ X e. S ) -> ( F ` X ) R ( G ` X ) ) |
| 18 |
|
simp1 |
|- ( ( ph /\ F oR R G /\ X e. S ) -> ph ) |
| 19 |
|
inss1 |
|- ( A i^i B ) C_ A |
| 20 |
5 19
|
eqsstrri |
|- S C_ A |
| 21 |
|
simp3 |
|- ( ( ph /\ F oR R G /\ X e. S ) -> X e. S ) |
| 22 |
20 21
|
sselid |
|- ( ( ph /\ F oR R G /\ X e. S ) -> X e. A ) |
| 23 |
18 22 6
|
syl2anc |
|- ( ( ph /\ F oR R G /\ X e. S ) -> ( F ` X ) = C ) |
| 24 |
|
inss2 |
|- ( A i^i B ) C_ B |
| 25 |
5 24
|
eqsstrri |
|- S C_ B |
| 26 |
25 21
|
sselid |
|- ( ( ph /\ F oR R G /\ X e. S ) -> X e. B ) |
| 27 |
18 26 7
|
syl2anc |
|- ( ( ph /\ F oR R G /\ X e. S ) -> ( G ` X ) = D ) |
| 28 |
17 23 27
|
3brtr3d |
|- ( ( ph /\ F oR R G /\ X e. S ) -> C R D ) |