Step |
Hyp |
Ref |
Expression |
1 |
|
ofun.a |
|- ( ph -> A Fn M ) |
2 |
|
ofun.b |
|- ( ph -> B Fn M ) |
3 |
|
ofun.c |
|- ( ph -> C Fn N ) |
4 |
|
ofun.d |
|- ( ph -> D Fn N ) |
5 |
|
ofun.m |
|- ( ph -> M e. V ) |
6 |
|
ofun.n |
|- ( ph -> N e. W ) |
7 |
|
ofun.1 |
|- ( ph -> ( M i^i N ) = (/) ) |
8 |
1 3 7
|
fnund |
|- ( ph -> ( A u. C ) Fn ( M u. N ) ) |
9 |
2 4 7
|
fnund |
|- ( ph -> ( B u. D ) Fn ( M u. N ) ) |
10 |
5 6
|
unexd |
|- ( ph -> ( M u. N ) e. _V ) |
11 |
|
inidm |
|- ( ( M u. N ) i^i ( M u. N ) ) = ( M u. N ) |
12 |
8 9 10 10 11
|
offn |
|- ( ph -> ( ( A u. C ) oF R ( B u. D ) ) Fn ( M u. N ) ) |
13 |
|
inidm |
|- ( M i^i M ) = M |
14 |
1 2 5 5 13
|
offn |
|- ( ph -> ( A oF R B ) Fn M ) |
15 |
|
inidm |
|- ( N i^i N ) = N |
16 |
3 4 6 6 15
|
offn |
|- ( ph -> ( C oF R D ) Fn N ) |
17 |
14 16 7
|
fnund |
|- ( ph -> ( ( A oF R B ) u. ( C oF R D ) ) Fn ( M u. N ) ) |
18 |
|
eqidd |
|- ( ( ph /\ x e. ( M u. N ) ) -> ( ( A u. C ) ` x ) = ( ( A u. C ) ` x ) ) |
19 |
|
eqidd |
|- ( ( ph /\ x e. ( M u. N ) ) -> ( ( B u. D ) ` x ) = ( ( B u. D ) ` x ) ) |
20 |
8 9 10 10 11 18 19
|
ofval |
|- ( ( ph /\ x e. ( M u. N ) ) -> ( ( ( A u. C ) oF R ( B u. D ) ) ` x ) = ( ( ( A u. C ) ` x ) R ( ( B u. D ) ` x ) ) ) |
21 |
|
elun |
|- ( x e. ( M u. N ) <-> ( x e. M \/ x e. N ) ) |
22 |
|
eqidd |
|- ( ( ph /\ x e. M ) -> ( A ` x ) = ( A ` x ) ) |
23 |
|
eqidd |
|- ( ( ph /\ x e. M ) -> ( B ` x ) = ( B ` x ) ) |
24 |
1 2 5 5 13 22 23
|
ofval |
|- ( ( ph /\ x e. M ) -> ( ( A oF R B ) ` x ) = ( ( A ` x ) R ( B ` x ) ) ) |
25 |
14
|
adantr |
|- ( ( ph /\ x e. M ) -> ( A oF R B ) Fn M ) |
26 |
16
|
adantr |
|- ( ( ph /\ x e. M ) -> ( C oF R D ) Fn N ) |
27 |
7
|
adantr |
|- ( ( ph /\ x e. M ) -> ( M i^i N ) = (/) ) |
28 |
|
simpr |
|- ( ( ph /\ x e. M ) -> x e. M ) |
29 |
25 26 27 28
|
fvun1d |
|- ( ( ph /\ x e. M ) -> ( ( ( A oF R B ) u. ( C oF R D ) ) ` x ) = ( ( A oF R B ) ` x ) ) |
30 |
1
|
adantr |
|- ( ( ph /\ x e. M ) -> A Fn M ) |
31 |
3
|
adantr |
|- ( ( ph /\ x e. M ) -> C Fn N ) |
32 |
30 31 27 28
|
fvun1d |
|- ( ( ph /\ x e. M ) -> ( ( A u. C ) ` x ) = ( A ` x ) ) |
33 |
2
|
adantr |
|- ( ( ph /\ x e. M ) -> B Fn M ) |
34 |
4
|
adantr |
|- ( ( ph /\ x e. M ) -> D Fn N ) |
35 |
33 34 27 28
|
fvun1d |
|- ( ( ph /\ x e. M ) -> ( ( B u. D ) ` x ) = ( B ` x ) ) |
36 |
32 35
|
oveq12d |
|- ( ( ph /\ x e. M ) -> ( ( ( A u. C ) ` x ) R ( ( B u. D ) ` x ) ) = ( ( A ` x ) R ( B ` x ) ) ) |
37 |
24 29 36
|
3eqtr4rd |
|- ( ( ph /\ x e. M ) -> ( ( ( A u. C ) ` x ) R ( ( B u. D ) ` x ) ) = ( ( ( A oF R B ) u. ( C oF R D ) ) ` x ) ) |
38 |
|
eqidd |
|- ( ( ph /\ x e. N ) -> ( C ` x ) = ( C ` x ) ) |
39 |
|
eqidd |
|- ( ( ph /\ x e. N ) -> ( D ` x ) = ( D ` x ) ) |
40 |
3 4 6 6 15 38 39
|
ofval |
|- ( ( ph /\ x e. N ) -> ( ( C oF R D ) ` x ) = ( ( C ` x ) R ( D ` x ) ) ) |
41 |
14
|
adantr |
|- ( ( ph /\ x e. N ) -> ( A oF R B ) Fn M ) |
42 |
16
|
adantr |
|- ( ( ph /\ x e. N ) -> ( C oF R D ) Fn N ) |
43 |
7
|
adantr |
|- ( ( ph /\ x e. N ) -> ( M i^i N ) = (/) ) |
44 |
|
simpr |
|- ( ( ph /\ x e. N ) -> x e. N ) |
45 |
41 42 43 44
|
fvun2d |
|- ( ( ph /\ x e. N ) -> ( ( ( A oF R B ) u. ( C oF R D ) ) ` x ) = ( ( C oF R D ) ` x ) ) |
46 |
1
|
adantr |
|- ( ( ph /\ x e. N ) -> A Fn M ) |
47 |
3
|
adantr |
|- ( ( ph /\ x e. N ) -> C Fn N ) |
48 |
46 47 43 44
|
fvun2d |
|- ( ( ph /\ x e. N ) -> ( ( A u. C ) ` x ) = ( C ` x ) ) |
49 |
2
|
adantr |
|- ( ( ph /\ x e. N ) -> B Fn M ) |
50 |
4
|
adantr |
|- ( ( ph /\ x e. N ) -> D Fn N ) |
51 |
49 50 43 44
|
fvun2d |
|- ( ( ph /\ x e. N ) -> ( ( B u. D ) ` x ) = ( D ` x ) ) |
52 |
48 51
|
oveq12d |
|- ( ( ph /\ x e. N ) -> ( ( ( A u. C ) ` x ) R ( ( B u. D ) ` x ) ) = ( ( C ` x ) R ( D ` x ) ) ) |
53 |
40 45 52
|
3eqtr4rd |
|- ( ( ph /\ x e. N ) -> ( ( ( A u. C ) ` x ) R ( ( B u. D ) ` x ) ) = ( ( ( A oF R B ) u. ( C oF R D ) ) ` x ) ) |
54 |
37 53
|
jaodan |
|- ( ( ph /\ ( x e. M \/ x e. N ) ) -> ( ( ( A u. C ) ` x ) R ( ( B u. D ) ` x ) ) = ( ( ( A oF R B ) u. ( C oF R D ) ) ` x ) ) |
55 |
21 54
|
sylan2b |
|- ( ( ph /\ x e. ( M u. N ) ) -> ( ( ( A u. C ) ` x ) R ( ( B u. D ) ` x ) ) = ( ( ( A oF R B ) u. ( C oF R D ) ) ` x ) ) |
56 |
20 55
|
eqtrd |
|- ( ( ph /\ x e. ( M u. N ) ) -> ( ( ( A u. C ) oF R ( B u. D ) ) ` x ) = ( ( ( A oF R B ) u. ( C oF R D ) ) ` x ) ) |
57 |
12 17 56
|
eqfnfvd |
|- ( ph -> ( ( A u. C ) oF R ( B u. D ) ) = ( ( A oF R B ) u. ( C oF R D ) ) ) |