Step |
Hyp |
Ref |
Expression |
1 |
|
ofun.a |
⊢ ( 𝜑 → 𝐴 Fn 𝑀 ) |
2 |
|
ofun.b |
⊢ ( 𝜑 → 𝐵 Fn 𝑀 ) |
3 |
|
ofun.c |
⊢ ( 𝜑 → 𝐶 Fn 𝑁 ) |
4 |
|
ofun.d |
⊢ ( 𝜑 → 𝐷 Fn 𝑁 ) |
5 |
|
ofun.m |
⊢ ( 𝜑 → 𝑀 ∈ 𝑉 ) |
6 |
|
ofun.n |
⊢ ( 𝜑 → 𝑁 ∈ 𝑊 ) |
7 |
|
ofun.1 |
⊢ ( 𝜑 → ( 𝑀 ∩ 𝑁 ) = ∅ ) |
8 |
1 3 7
|
fnund |
⊢ ( 𝜑 → ( 𝐴 ∪ 𝐶 ) Fn ( 𝑀 ∪ 𝑁 ) ) |
9 |
2 4 7
|
fnund |
⊢ ( 𝜑 → ( 𝐵 ∪ 𝐷 ) Fn ( 𝑀 ∪ 𝑁 ) ) |
10 |
5 6
|
unexd |
⊢ ( 𝜑 → ( 𝑀 ∪ 𝑁 ) ∈ V ) |
11 |
|
inidm |
⊢ ( ( 𝑀 ∪ 𝑁 ) ∩ ( 𝑀 ∪ 𝑁 ) ) = ( 𝑀 ∪ 𝑁 ) |
12 |
8 9 10 10 11
|
offn |
⊢ ( 𝜑 → ( ( 𝐴 ∪ 𝐶 ) ∘f 𝑅 ( 𝐵 ∪ 𝐷 ) ) Fn ( 𝑀 ∪ 𝑁 ) ) |
13 |
|
inidm |
⊢ ( 𝑀 ∩ 𝑀 ) = 𝑀 |
14 |
1 2 5 5 13
|
offn |
⊢ ( 𝜑 → ( 𝐴 ∘f 𝑅 𝐵 ) Fn 𝑀 ) |
15 |
|
inidm |
⊢ ( 𝑁 ∩ 𝑁 ) = 𝑁 |
16 |
3 4 6 6 15
|
offn |
⊢ ( 𝜑 → ( 𝐶 ∘f 𝑅 𝐷 ) Fn 𝑁 ) |
17 |
14 16 7
|
fnund |
⊢ ( 𝜑 → ( ( 𝐴 ∘f 𝑅 𝐵 ) ∪ ( 𝐶 ∘f 𝑅 𝐷 ) ) Fn ( 𝑀 ∪ 𝑁 ) ) |
18 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ∪ 𝑁 ) ) → ( ( 𝐴 ∪ 𝐶 ) ‘ 𝑥 ) = ( ( 𝐴 ∪ 𝐶 ) ‘ 𝑥 ) ) |
19 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ∪ 𝑁 ) ) → ( ( 𝐵 ∪ 𝐷 ) ‘ 𝑥 ) = ( ( 𝐵 ∪ 𝐷 ) ‘ 𝑥 ) ) |
20 |
8 9 10 10 11 18 19
|
ofval |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ∪ 𝑁 ) ) → ( ( ( 𝐴 ∪ 𝐶 ) ∘f 𝑅 ( 𝐵 ∪ 𝐷 ) ) ‘ 𝑥 ) = ( ( ( 𝐴 ∪ 𝐶 ) ‘ 𝑥 ) 𝑅 ( ( 𝐵 ∪ 𝐷 ) ‘ 𝑥 ) ) ) |
21 |
|
elun |
⊢ ( 𝑥 ∈ ( 𝑀 ∪ 𝑁 ) ↔ ( 𝑥 ∈ 𝑀 ∨ 𝑥 ∈ 𝑁 ) ) |
22 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ 𝑥 ) ) |
23 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → ( 𝐵 ‘ 𝑥 ) = ( 𝐵 ‘ 𝑥 ) ) |
24 |
1 2 5 5 13 22 23
|
ofval |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → ( ( 𝐴 ∘f 𝑅 𝐵 ) ‘ 𝑥 ) = ( ( 𝐴 ‘ 𝑥 ) 𝑅 ( 𝐵 ‘ 𝑥 ) ) ) |
25 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → ( 𝐴 ∘f 𝑅 𝐵 ) Fn 𝑀 ) |
26 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → ( 𝐶 ∘f 𝑅 𝐷 ) Fn 𝑁 ) |
27 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → ( 𝑀 ∩ 𝑁 ) = ∅ ) |
28 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → 𝑥 ∈ 𝑀 ) |
29 |
25 26 27 28
|
fvun1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → ( ( ( 𝐴 ∘f 𝑅 𝐵 ) ∪ ( 𝐶 ∘f 𝑅 𝐷 ) ) ‘ 𝑥 ) = ( ( 𝐴 ∘f 𝑅 𝐵 ) ‘ 𝑥 ) ) |
30 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → 𝐴 Fn 𝑀 ) |
31 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → 𝐶 Fn 𝑁 ) |
32 |
30 31 27 28
|
fvun1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → ( ( 𝐴 ∪ 𝐶 ) ‘ 𝑥 ) = ( 𝐴 ‘ 𝑥 ) ) |
33 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → 𝐵 Fn 𝑀 ) |
34 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → 𝐷 Fn 𝑁 ) |
35 |
33 34 27 28
|
fvun1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → ( ( 𝐵 ∪ 𝐷 ) ‘ 𝑥 ) = ( 𝐵 ‘ 𝑥 ) ) |
36 |
32 35
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → ( ( ( 𝐴 ∪ 𝐶 ) ‘ 𝑥 ) 𝑅 ( ( 𝐵 ∪ 𝐷 ) ‘ 𝑥 ) ) = ( ( 𝐴 ‘ 𝑥 ) 𝑅 ( 𝐵 ‘ 𝑥 ) ) ) |
37 |
24 29 36
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → ( ( ( 𝐴 ∪ 𝐶 ) ‘ 𝑥 ) 𝑅 ( ( 𝐵 ∪ 𝐷 ) ‘ 𝑥 ) ) = ( ( ( 𝐴 ∘f 𝑅 𝐵 ) ∪ ( 𝐶 ∘f 𝑅 𝐷 ) ) ‘ 𝑥 ) ) |
38 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → ( 𝐶 ‘ 𝑥 ) = ( 𝐶 ‘ 𝑥 ) ) |
39 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → ( 𝐷 ‘ 𝑥 ) = ( 𝐷 ‘ 𝑥 ) ) |
40 |
3 4 6 6 15 38 39
|
ofval |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → ( ( 𝐶 ∘f 𝑅 𝐷 ) ‘ 𝑥 ) = ( ( 𝐶 ‘ 𝑥 ) 𝑅 ( 𝐷 ‘ 𝑥 ) ) ) |
41 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → ( 𝐴 ∘f 𝑅 𝐵 ) Fn 𝑀 ) |
42 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → ( 𝐶 ∘f 𝑅 𝐷 ) Fn 𝑁 ) |
43 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → ( 𝑀 ∩ 𝑁 ) = ∅ ) |
44 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → 𝑥 ∈ 𝑁 ) |
45 |
41 42 43 44
|
fvun2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → ( ( ( 𝐴 ∘f 𝑅 𝐵 ) ∪ ( 𝐶 ∘f 𝑅 𝐷 ) ) ‘ 𝑥 ) = ( ( 𝐶 ∘f 𝑅 𝐷 ) ‘ 𝑥 ) ) |
46 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → 𝐴 Fn 𝑀 ) |
47 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → 𝐶 Fn 𝑁 ) |
48 |
46 47 43 44
|
fvun2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → ( ( 𝐴 ∪ 𝐶 ) ‘ 𝑥 ) = ( 𝐶 ‘ 𝑥 ) ) |
49 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → 𝐵 Fn 𝑀 ) |
50 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → 𝐷 Fn 𝑁 ) |
51 |
49 50 43 44
|
fvun2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → ( ( 𝐵 ∪ 𝐷 ) ‘ 𝑥 ) = ( 𝐷 ‘ 𝑥 ) ) |
52 |
48 51
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → ( ( ( 𝐴 ∪ 𝐶 ) ‘ 𝑥 ) 𝑅 ( ( 𝐵 ∪ 𝐷 ) ‘ 𝑥 ) ) = ( ( 𝐶 ‘ 𝑥 ) 𝑅 ( 𝐷 ‘ 𝑥 ) ) ) |
53 |
40 45 52
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → ( ( ( 𝐴 ∪ 𝐶 ) ‘ 𝑥 ) 𝑅 ( ( 𝐵 ∪ 𝐷 ) ‘ 𝑥 ) ) = ( ( ( 𝐴 ∘f 𝑅 𝐵 ) ∪ ( 𝐶 ∘f 𝑅 𝐷 ) ) ‘ 𝑥 ) ) |
54 |
37 53
|
jaodan |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑀 ∨ 𝑥 ∈ 𝑁 ) ) → ( ( ( 𝐴 ∪ 𝐶 ) ‘ 𝑥 ) 𝑅 ( ( 𝐵 ∪ 𝐷 ) ‘ 𝑥 ) ) = ( ( ( 𝐴 ∘f 𝑅 𝐵 ) ∪ ( 𝐶 ∘f 𝑅 𝐷 ) ) ‘ 𝑥 ) ) |
55 |
21 54
|
sylan2b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ∪ 𝑁 ) ) → ( ( ( 𝐴 ∪ 𝐶 ) ‘ 𝑥 ) 𝑅 ( ( 𝐵 ∪ 𝐷 ) ‘ 𝑥 ) ) = ( ( ( 𝐴 ∘f 𝑅 𝐵 ) ∪ ( 𝐶 ∘f 𝑅 𝐷 ) ) ‘ 𝑥 ) ) |
56 |
20 55
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ∪ 𝑁 ) ) → ( ( ( 𝐴 ∪ 𝐶 ) ∘f 𝑅 ( 𝐵 ∪ 𝐷 ) ) ‘ 𝑥 ) = ( ( ( 𝐴 ∘f 𝑅 𝐵 ) ∪ ( 𝐶 ∘f 𝑅 𝐷 ) ) ‘ 𝑥 ) ) |
57 |
12 17 56
|
eqfnfvd |
⊢ ( 𝜑 → ( ( 𝐴 ∪ 𝐶 ) ∘f 𝑅 ( 𝐵 ∪ 𝐷 ) ) = ( ( 𝐴 ∘f 𝑅 𝐵 ) ∪ ( 𝐶 ∘f 𝑅 𝐷 ) ) ) |