Step |
Hyp |
Ref |
Expression |
1 |
|
ofco.1 |
|- ( ph -> F Fn A ) |
2 |
|
ofco.2 |
|- ( ph -> G Fn B ) |
3 |
|
ofco.3 |
|- ( ph -> H : D --> C ) |
4 |
|
ofco.4 |
|- ( ph -> A e. V ) |
5 |
|
ofco.5 |
|- ( ph -> B e. W ) |
6 |
|
ofco.6 |
|- ( ph -> D e. X ) |
7 |
|
ofco.7 |
|- ( A i^i B ) = C |
8 |
3
|
ffvelrnda |
|- ( ( ph /\ x e. D ) -> ( H ` x ) e. C ) |
9 |
3
|
feqmptd |
|- ( ph -> H = ( x e. D |-> ( H ` x ) ) ) |
10 |
|
eqidd |
|- ( ( ph /\ y e. A ) -> ( F ` y ) = ( F ` y ) ) |
11 |
|
eqidd |
|- ( ( ph /\ y e. B ) -> ( G ` y ) = ( G ` y ) ) |
12 |
1 2 4 5 7 10 11
|
offval |
|- ( ph -> ( F oF R G ) = ( y e. C |-> ( ( F ` y ) R ( G ` y ) ) ) ) |
13 |
|
fveq2 |
|- ( y = ( H ` x ) -> ( F ` y ) = ( F ` ( H ` x ) ) ) |
14 |
|
fveq2 |
|- ( y = ( H ` x ) -> ( G ` y ) = ( G ` ( H ` x ) ) ) |
15 |
13 14
|
oveq12d |
|- ( y = ( H ` x ) -> ( ( F ` y ) R ( G ` y ) ) = ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) ) |
16 |
8 9 12 15
|
fmptco |
|- ( ph -> ( ( F oF R G ) o. H ) = ( x e. D |-> ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) ) ) |
17 |
|
inss1 |
|- ( A i^i B ) C_ A |
18 |
7 17
|
eqsstrri |
|- C C_ A |
19 |
|
fss |
|- ( ( H : D --> C /\ C C_ A ) -> H : D --> A ) |
20 |
3 18 19
|
sylancl |
|- ( ph -> H : D --> A ) |
21 |
|
fnfco |
|- ( ( F Fn A /\ H : D --> A ) -> ( F o. H ) Fn D ) |
22 |
1 20 21
|
syl2anc |
|- ( ph -> ( F o. H ) Fn D ) |
23 |
|
inss2 |
|- ( A i^i B ) C_ B |
24 |
7 23
|
eqsstrri |
|- C C_ B |
25 |
|
fss |
|- ( ( H : D --> C /\ C C_ B ) -> H : D --> B ) |
26 |
3 24 25
|
sylancl |
|- ( ph -> H : D --> B ) |
27 |
|
fnfco |
|- ( ( G Fn B /\ H : D --> B ) -> ( G o. H ) Fn D ) |
28 |
2 26 27
|
syl2anc |
|- ( ph -> ( G o. H ) Fn D ) |
29 |
|
inidm |
|- ( D i^i D ) = D |
30 |
3
|
ffnd |
|- ( ph -> H Fn D ) |
31 |
|
fvco2 |
|- ( ( H Fn D /\ x e. D ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) ) |
32 |
30 31
|
sylan |
|- ( ( ph /\ x e. D ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) ) |
33 |
|
fvco2 |
|- ( ( H Fn D /\ x e. D ) -> ( ( G o. H ) ` x ) = ( G ` ( H ` x ) ) ) |
34 |
30 33
|
sylan |
|- ( ( ph /\ x e. D ) -> ( ( G o. H ) ` x ) = ( G ` ( H ` x ) ) ) |
35 |
22 28 6 6 29 32 34
|
offval |
|- ( ph -> ( ( F o. H ) oF R ( G o. H ) ) = ( x e. D |-> ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) ) ) |
36 |
16 35
|
eqtr4d |
|- ( ph -> ( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) ) |