Step |
Hyp |
Ref |
Expression |
1 |
|
off2.1 |
|- ( ( ph /\ ( x e. S /\ y e. T ) ) -> ( x R y ) e. U ) |
2 |
|
off2.2 |
|- ( ph -> F : A --> S ) |
3 |
|
off2.3 |
|- ( ph -> G : B --> T ) |
4 |
|
off2.4 |
|- ( ph -> A e. V ) |
5 |
|
off2.5 |
|- ( ph -> B e. W ) |
6 |
|
off2.6 |
|- ( ph -> ( A i^i B ) = C ) |
7 |
2
|
ffnd |
|- ( ph -> F Fn A ) |
8 |
3
|
ffnd |
|- ( ph -> G Fn B ) |
9 |
|
eqid |
|- ( A i^i B ) = ( A i^i B ) |
10 |
|
eqidd |
|- ( ( ph /\ z e. A ) -> ( F ` z ) = ( F ` z ) ) |
11 |
|
eqidd |
|- ( ( ph /\ z e. B ) -> ( G ` z ) = ( G ` z ) ) |
12 |
7 8 4 5 9 10 11
|
offval |
|- ( ph -> ( F oF R G ) = ( z e. ( A i^i B ) |-> ( ( F ` z ) R ( G ` z ) ) ) ) |
13 |
6
|
mpteq1d |
|- ( ph -> ( z e. ( A i^i B ) |-> ( ( F ` z ) R ( G ` z ) ) ) = ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) ) |
14 |
12 13
|
eqtrd |
|- ( ph -> ( F oF R G ) = ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) ) |
15 |
2
|
adantr |
|- ( ( ph /\ z e. C ) -> F : A --> S ) |
16 |
|
inss1 |
|- ( A i^i B ) C_ A |
17 |
6 16
|
eqsstrrdi |
|- ( ph -> C C_ A ) |
18 |
17
|
sselda |
|- ( ( ph /\ z e. C ) -> z e. A ) |
19 |
15 18
|
ffvelrnd |
|- ( ( ph /\ z e. C ) -> ( F ` z ) e. S ) |
20 |
3
|
adantr |
|- ( ( ph /\ z e. C ) -> G : B --> T ) |
21 |
|
inss2 |
|- ( A i^i B ) C_ B |
22 |
6 21
|
eqsstrrdi |
|- ( ph -> C C_ B ) |
23 |
22
|
sselda |
|- ( ( ph /\ z e. C ) -> z e. B ) |
24 |
20 23
|
ffvelrnd |
|- ( ( ph /\ z e. C ) -> ( G ` z ) e. T ) |
25 |
1
|
ralrimivva |
|- ( ph -> A. x e. S A. y e. T ( x R y ) e. U ) |
26 |
25
|
adantr |
|- ( ( ph /\ z e. C ) -> A. x e. S A. y e. T ( x R y ) e. U ) |
27 |
|
ovrspc2v |
|- ( ( ( ( F ` z ) e. S /\ ( G ` z ) e. T ) /\ A. x e. S A. y e. T ( x R y ) e. U ) -> ( ( F ` z ) R ( G ` z ) ) e. U ) |
28 |
19 24 26 27
|
syl21anc |
|- ( ( ph /\ z e. C ) -> ( ( F ` z ) R ( G ` z ) ) e. U ) |
29 |
14 28
|
fmpt3d |
|- ( ph -> ( F oF R G ) : C --> U ) |