Step |
Hyp |
Ref |
Expression |
1 |
|
mrsubco.s |
⊢ 𝑆 = ( mRSubst ‘ 𝑇 ) |
2 |
|
mrsubvrs.v |
⊢ 𝑉 = ( mVR ‘ 𝑇 ) |
3 |
|
mrsubvrs.r |
⊢ 𝑅 = ( mREx ‘ 𝑇 ) |
4 |
|
n0i |
⊢ ( 𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅ ) |
5 |
1
|
rnfvprc |
⊢ ( ¬ 𝑇 ∈ V → ran 𝑆 = ∅ ) |
6 |
4 5
|
nsyl2 |
⊢ ( 𝐹 ∈ ran 𝑆 → 𝑇 ∈ V ) |
7 |
|
eqid |
⊢ ( mCN ‘ 𝑇 ) = ( mCN ‘ 𝑇 ) |
8 |
7 2 3
|
mrexval |
⊢ ( 𝑇 ∈ V → 𝑅 = Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) |
9 |
6 8
|
syl |
⊢ ( 𝐹 ∈ ran 𝑆 → 𝑅 = Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) |
10 |
9
|
eleq2d |
⊢ ( 𝐹 ∈ ran 𝑆 → ( 𝑋 ∈ 𝑅 ↔ 𝑋 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) |
11 |
|
fveq2 |
⊢ ( 𝑣 = ∅ → ( 𝐹 ‘ 𝑣 ) = ( 𝐹 ‘ ∅ ) ) |
12 |
11
|
rneqd |
⊢ ( 𝑣 = ∅ → ran ( 𝐹 ‘ 𝑣 ) = ran ( 𝐹 ‘ ∅ ) ) |
13 |
12
|
ineq1d |
⊢ ( 𝑣 = ∅ → ( ran ( 𝐹 ‘ 𝑣 ) ∩ 𝑉 ) = ( ran ( 𝐹 ‘ ∅ ) ∩ 𝑉 ) ) |
14 |
|
rneq |
⊢ ( 𝑣 = ∅ → ran 𝑣 = ran ∅ ) |
15 |
|
rn0 |
⊢ ran ∅ = ∅ |
16 |
14 15
|
eqtrdi |
⊢ ( 𝑣 = ∅ → ran 𝑣 = ∅ ) |
17 |
16
|
ineq1d |
⊢ ( 𝑣 = ∅ → ( ran 𝑣 ∩ 𝑉 ) = ( ∅ ∩ 𝑉 ) ) |
18 |
|
0in |
⊢ ( ∅ ∩ 𝑉 ) = ∅ |
19 |
17 18
|
eqtrdi |
⊢ ( 𝑣 = ∅ → ( ran 𝑣 ∩ 𝑉 ) = ∅ ) |
20 |
19
|
iuneq1d |
⊢ ( 𝑣 = ∅ → ∪ 𝑥 ∈ ( ran 𝑣 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ∅ ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) |
21 |
|
0iun |
⊢ ∪ 𝑥 ∈ ∅ ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ∅ |
22 |
20 21
|
eqtrdi |
⊢ ( 𝑣 = ∅ → ∪ 𝑥 ∈ ( ran 𝑣 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ∅ ) |
23 |
13 22
|
eqeq12d |
⊢ ( 𝑣 = ∅ → ( ( ran ( 𝐹 ‘ 𝑣 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑣 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ↔ ( ran ( 𝐹 ‘ ∅ ) ∩ 𝑉 ) = ∅ ) ) |
24 |
23
|
imbi2d |
⊢ ( 𝑣 = ∅ → ( ( 𝐹 ∈ ran 𝑆 → ( ran ( 𝐹 ‘ 𝑣 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑣 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ↔ ( 𝐹 ∈ ran 𝑆 → ( ran ( 𝐹 ‘ ∅ ) ∩ 𝑉 ) = ∅ ) ) ) |
25 |
|
fveq2 |
⊢ ( 𝑣 = 𝑦 → ( 𝐹 ‘ 𝑣 ) = ( 𝐹 ‘ 𝑦 ) ) |
26 |
25
|
rneqd |
⊢ ( 𝑣 = 𝑦 → ran ( 𝐹 ‘ 𝑣 ) = ran ( 𝐹 ‘ 𝑦 ) ) |
27 |
26
|
ineq1d |
⊢ ( 𝑣 = 𝑦 → ( ran ( 𝐹 ‘ 𝑣 ) ∩ 𝑉 ) = ( ran ( 𝐹 ‘ 𝑦 ) ∩ 𝑉 ) ) |
28 |
|
rneq |
⊢ ( 𝑣 = 𝑦 → ran 𝑣 = ran 𝑦 ) |
29 |
28
|
ineq1d |
⊢ ( 𝑣 = 𝑦 → ( ran 𝑣 ∩ 𝑉 ) = ( ran 𝑦 ∩ 𝑉 ) ) |
30 |
29
|
iuneq1d |
⊢ ( 𝑣 = 𝑦 → ∪ 𝑥 ∈ ( ran 𝑣 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑦 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) |
31 |
27 30
|
eqeq12d |
⊢ ( 𝑣 = 𝑦 → ( ( ran ( 𝐹 ‘ 𝑣 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑣 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ↔ ( ran ( 𝐹 ‘ 𝑦 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑦 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ) |
32 |
31
|
imbi2d |
⊢ ( 𝑣 = 𝑦 → ( ( 𝐹 ∈ ran 𝑆 → ( ran ( 𝐹 ‘ 𝑣 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑣 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ↔ ( 𝐹 ∈ ran 𝑆 → ( ran ( 𝐹 ‘ 𝑦 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑦 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ) ) |
33 |
|
fveq2 |
⊢ ( 𝑣 = ( 𝑦 ++ 〈“ 𝑧 ”〉 ) → ( 𝐹 ‘ 𝑣 ) = ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) ) |
34 |
33
|
rneqd |
⊢ ( 𝑣 = ( 𝑦 ++ 〈“ 𝑧 ”〉 ) → ran ( 𝐹 ‘ 𝑣 ) = ran ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) ) |
35 |
34
|
ineq1d |
⊢ ( 𝑣 = ( 𝑦 ++ 〈“ 𝑧 ”〉 ) → ( ran ( 𝐹 ‘ 𝑣 ) ∩ 𝑉 ) = ( ran ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) ∩ 𝑉 ) ) |
36 |
|
rneq |
⊢ ( 𝑣 = ( 𝑦 ++ 〈“ 𝑧 ”〉 ) → ran 𝑣 = ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) |
37 |
36
|
ineq1d |
⊢ ( 𝑣 = ( 𝑦 ++ 〈“ 𝑧 ”〉 ) → ( ran 𝑣 ∩ 𝑉 ) = ( ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ) |
38 |
37
|
iuneq1d |
⊢ ( 𝑣 = ( 𝑦 ++ 〈“ 𝑧 ”〉 ) → ∪ 𝑥 ∈ ( ran 𝑣 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) |
39 |
35 38
|
eqeq12d |
⊢ ( 𝑣 = ( 𝑦 ++ 〈“ 𝑧 ”〉 ) → ( ( ran ( 𝐹 ‘ 𝑣 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑣 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ↔ ( ran ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ) |
40 |
39
|
imbi2d |
⊢ ( 𝑣 = ( 𝑦 ++ 〈“ 𝑧 ”〉 ) → ( ( 𝐹 ∈ ran 𝑆 → ( ran ( 𝐹 ‘ 𝑣 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑣 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ↔ ( 𝐹 ∈ ran 𝑆 → ( ran ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ) ) |
41 |
|
fveq2 |
⊢ ( 𝑣 = 𝑋 → ( 𝐹 ‘ 𝑣 ) = ( 𝐹 ‘ 𝑋 ) ) |
42 |
41
|
rneqd |
⊢ ( 𝑣 = 𝑋 → ran ( 𝐹 ‘ 𝑣 ) = ran ( 𝐹 ‘ 𝑋 ) ) |
43 |
42
|
ineq1d |
⊢ ( 𝑣 = 𝑋 → ( ran ( 𝐹 ‘ 𝑣 ) ∩ 𝑉 ) = ( ran ( 𝐹 ‘ 𝑋 ) ∩ 𝑉 ) ) |
44 |
|
rneq |
⊢ ( 𝑣 = 𝑋 → ran 𝑣 = ran 𝑋 ) |
45 |
44
|
ineq1d |
⊢ ( 𝑣 = 𝑋 → ( ran 𝑣 ∩ 𝑉 ) = ( ran 𝑋 ∩ 𝑉 ) ) |
46 |
45
|
iuneq1d |
⊢ ( 𝑣 = 𝑋 → ∪ 𝑥 ∈ ( ran 𝑣 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑋 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) |
47 |
43 46
|
eqeq12d |
⊢ ( 𝑣 = 𝑋 → ( ( ran ( 𝐹 ‘ 𝑣 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑣 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ↔ ( ran ( 𝐹 ‘ 𝑋 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑋 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ) |
48 |
47
|
imbi2d |
⊢ ( 𝑣 = 𝑋 → ( ( 𝐹 ∈ ran 𝑆 → ( ran ( 𝐹 ‘ 𝑣 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑣 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ↔ ( 𝐹 ∈ ran 𝑆 → ( ran ( 𝐹 ‘ 𝑋 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑋 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ) ) |
49 |
1
|
mrsub0 |
⊢ ( 𝐹 ∈ ran 𝑆 → ( 𝐹 ‘ ∅ ) = ∅ ) |
50 |
49
|
rneqd |
⊢ ( 𝐹 ∈ ran 𝑆 → ran ( 𝐹 ‘ ∅ ) = ran ∅ ) |
51 |
50 15
|
eqtrdi |
⊢ ( 𝐹 ∈ ran 𝑆 → ran ( 𝐹 ‘ ∅ ) = ∅ ) |
52 |
51
|
ineq1d |
⊢ ( 𝐹 ∈ ran 𝑆 → ( ran ( 𝐹 ‘ ∅ ) ∩ 𝑉 ) = ( ∅ ∩ 𝑉 ) ) |
53 |
52 18
|
eqtrdi |
⊢ ( 𝐹 ∈ ran 𝑆 → ( ran ( 𝐹 ‘ ∅ ) ∩ 𝑉 ) = ∅ ) |
54 |
|
uneq1 |
⊢ ( ( ran ( 𝐹 ‘ 𝑦 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑦 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) → ( ( ran ( 𝐹 ‘ 𝑦 ) ∩ 𝑉 ) ∪ ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ) = ( ∪ 𝑥 ∈ ( ran 𝑦 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ∪ ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ) ) |
55 |
|
simpl |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → 𝐹 ∈ ran 𝑆 ) |
56 |
|
simprl |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) |
57 |
9
|
adantr |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → 𝑅 = Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) |
58 |
56 57
|
eleqtrrd |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → 𝑦 ∈ 𝑅 ) |
59 |
|
simprr |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) |
60 |
59
|
s1cld |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → 〈“ 𝑧 ”〉 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) |
61 |
60 57
|
eleqtrrd |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → 〈“ 𝑧 ”〉 ∈ 𝑅 ) |
62 |
1 3
|
mrsubccat |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ 𝑦 ∈ 𝑅 ∧ 〈“ 𝑧 ”〉 ∈ 𝑅 ) → ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) = ( ( 𝐹 ‘ 𝑦 ) ++ ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ) ) |
63 |
55 58 61 62
|
syl3anc |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) = ( ( 𝐹 ‘ 𝑦 ) ++ ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ) ) |
64 |
63
|
rneqd |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ran ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) = ran ( ( 𝐹 ‘ 𝑦 ) ++ ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ) ) |
65 |
1 3
|
mrsubf |
⊢ ( 𝐹 ∈ ran 𝑆 → 𝐹 : 𝑅 ⟶ 𝑅 ) |
66 |
65
|
adantr |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → 𝐹 : 𝑅 ⟶ 𝑅 ) |
67 |
66 58
|
ffvelrnd |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑅 ) |
68 |
67 57
|
eleqtrd |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ( 𝐹 ‘ 𝑦 ) ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) |
69 |
66 61
|
ffvelrnd |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∈ 𝑅 ) |
70 |
69 57
|
eleqtrd |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) |
71 |
|
ccatrn |
⊢ ( ( ( 𝐹 ‘ 𝑦 ) ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) → ran ( ( 𝐹 ‘ 𝑦 ) ++ ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ) = ( ran ( 𝐹 ‘ 𝑦 ) ∪ ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ) ) |
72 |
68 70 71
|
syl2anc |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ran ( ( 𝐹 ‘ 𝑦 ) ++ ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ) = ( ran ( 𝐹 ‘ 𝑦 ) ∪ ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ) ) |
73 |
64 72
|
eqtrd |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ran ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) = ( ran ( 𝐹 ‘ 𝑦 ) ∪ ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ) ) |
74 |
73
|
ineq1d |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ( ran ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) ∩ 𝑉 ) = ( ( ran ( 𝐹 ‘ 𝑦 ) ∪ ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ) ∩ 𝑉 ) ) |
75 |
|
indir |
⊢ ( ( ran ( 𝐹 ‘ 𝑦 ) ∪ ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ) ∩ 𝑉 ) = ( ( ran ( 𝐹 ‘ 𝑦 ) ∩ 𝑉 ) ∪ ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ) |
76 |
74 75
|
eqtrdi |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ( ran ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) ∩ 𝑉 ) = ( ( ran ( 𝐹 ‘ 𝑦 ) ∩ 𝑉 ) ∪ ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ) ) |
77 |
|
ccatrn |
⊢ ( ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 〈“ 𝑧 ”〉 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) → ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) = ( ran 𝑦 ∪ ran 〈“ 𝑧 ”〉 ) ) |
78 |
56 60 77
|
syl2anc |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) = ( ran 𝑦 ∪ ran 〈“ 𝑧 ”〉 ) ) |
79 |
|
s1rn |
⊢ ( 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) → ran 〈“ 𝑧 ”〉 = { 𝑧 } ) |
80 |
79
|
ad2antll |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ran 〈“ 𝑧 ”〉 = { 𝑧 } ) |
81 |
80
|
uneq2d |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ( ran 𝑦 ∪ ran 〈“ 𝑧 ”〉 ) = ( ran 𝑦 ∪ { 𝑧 } ) ) |
82 |
78 81
|
eqtrd |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) = ( ran 𝑦 ∪ { 𝑧 } ) ) |
83 |
82
|
ineq1d |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ( ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) = ( ( ran 𝑦 ∪ { 𝑧 } ) ∩ 𝑉 ) ) |
84 |
|
indir |
⊢ ( ( ran 𝑦 ∪ { 𝑧 } ) ∩ 𝑉 ) = ( ( ran 𝑦 ∩ 𝑉 ) ∪ ( { 𝑧 } ∩ 𝑉 ) ) |
85 |
83 84
|
eqtrdi |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ( ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) = ( ( ran 𝑦 ∩ 𝑉 ) ∪ ( { 𝑧 } ∩ 𝑉 ) ) ) |
86 |
85
|
iuneq1d |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ∪ 𝑥 ∈ ( ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ( ran 𝑦 ∩ 𝑉 ) ∪ ( { 𝑧 } ∩ 𝑉 ) ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) |
87 |
|
iunxun |
⊢ ∪ 𝑥 ∈ ( ( ran 𝑦 ∩ 𝑉 ) ∪ ( { 𝑧 } ∩ 𝑉 ) ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ( ∪ 𝑥 ∈ ( ran 𝑦 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ∪ ∪ 𝑥 ∈ ( { 𝑧 } ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) |
88 |
86 87
|
eqtrdi |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ∪ 𝑥 ∈ ( ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ( ∪ 𝑥 ∈ ( ran 𝑦 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ∪ ∪ 𝑥 ∈ ( { 𝑧 } ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ) |
89 |
|
simpr |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ 𝑧 ∈ 𝑉 ) → 𝑧 ∈ 𝑉 ) |
90 |
89
|
snssd |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ 𝑧 ∈ 𝑉 ) → { 𝑧 } ⊆ 𝑉 ) |
91 |
|
df-ss |
⊢ ( { 𝑧 } ⊆ 𝑉 ↔ ( { 𝑧 } ∩ 𝑉 ) = { 𝑧 } ) |
92 |
90 91
|
sylib |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ 𝑧 ∈ 𝑉 ) → ( { 𝑧 } ∩ 𝑉 ) = { 𝑧 } ) |
93 |
92
|
iuneq1d |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ 𝑧 ∈ 𝑉 ) → ∪ 𝑥 ∈ ( { 𝑧 } ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ { 𝑧 } ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) |
94 |
|
vex |
⊢ 𝑧 ∈ V |
95 |
|
s1eq |
⊢ ( 𝑥 = 𝑧 → 〈“ 𝑥 ”〉 = 〈“ 𝑧 ”〉 ) |
96 |
95
|
fveq2d |
⊢ ( 𝑥 = 𝑧 → ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) = ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ) |
97 |
96
|
rneqd |
⊢ ( 𝑥 = 𝑧 → ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) = ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ) |
98 |
97
|
ineq1d |
⊢ ( 𝑥 = 𝑧 → ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ) |
99 |
94 98
|
iunxsn |
⊢ ∪ 𝑥 ∈ { 𝑧 } ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) |
100 |
93 99
|
eqtrdi |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ 𝑧 ∈ 𝑉 ) → ∪ 𝑥 ∈ ( { 𝑧 } ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ) |
101 |
|
incom |
⊢ ( { 𝑧 } ∩ 𝑉 ) = ( 𝑉 ∩ { 𝑧 } ) |
102 |
|
simpr |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ ¬ 𝑧 ∈ 𝑉 ) → ¬ 𝑧 ∈ 𝑉 ) |
103 |
|
disjsn |
⊢ ( ( 𝑉 ∩ { 𝑧 } ) = ∅ ↔ ¬ 𝑧 ∈ 𝑉 ) |
104 |
102 103
|
sylibr |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ ¬ 𝑧 ∈ 𝑉 ) → ( 𝑉 ∩ { 𝑧 } ) = ∅ ) |
105 |
101 104
|
syl5eq |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ ¬ 𝑧 ∈ 𝑉 ) → ( { 𝑧 } ∩ 𝑉 ) = ∅ ) |
106 |
105
|
iuneq1d |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ ¬ 𝑧 ∈ 𝑉 ) → ∪ 𝑥 ∈ ( { 𝑧 } ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ∅ ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) |
107 |
55
|
adantr |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ ¬ 𝑧 ∈ 𝑉 ) → 𝐹 ∈ ran 𝑆 ) |
108 |
|
eldif |
⊢ ( 𝑧 ∈ ( ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∖ 𝑉 ) ↔ ( 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ ¬ 𝑧 ∈ 𝑉 ) ) |
109 |
108
|
biimpri |
⊢ ( ( 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ ¬ 𝑧 ∈ 𝑉 ) → 𝑧 ∈ ( ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∖ 𝑉 ) ) |
110 |
59 109
|
sylan |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ ¬ 𝑧 ∈ 𝑉 ) → 𝑧 ∈ ( ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∖ 𝑉 ) ) |
111 |
|
difun2 |
⊢ ( ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∖ 𝑉 ) = ( ( mCN ‘ 𝑇 ) ∖ 𝑉 ) |
112 |
110 111
|
eleqtrdi |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ ¬ 𝑧 ∈ 𝑉 ) → 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∖ 𝑉 ) ) |
113 |
1 3 2 7
|
mrsubcn |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∖ 𝑉 ) ) → ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) = 〈“ 𝑧 ”〉 ) |
114 |
107 112 113
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ ¬ 𝑧 ∈ 𝑉 ) → ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) = 〈“ 𝑧 ”〉 ) |
115 |
114
|
rneqd |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ ¬ 𝑧 ∈ 𝑉 ) → ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) = ran 〈“ 𝑧 ”〉 ) |
116 |
80
|
adantr |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ ¬ 𝑧 ∈ 𝑉 ) → ran 〈“ 𝑧 ”〉 = { 𝑧 } ) |
117 |
115 116
|
eqtrd |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ ¬ 𝑧 ∈ 𝑉 ) → ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) = { 𝑧 } ) |
118 |
117
|
ineq1d |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ ¬ 𝑧 ∈ 𝑉 ) → ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) = ( { 𝑧 } ∩ 𝑉 ) ) |
119 |
118 105
|
eqtrd |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ ¬ 𝑧 ∈ 𝑉 ) → ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) = ∅ ) |
120 |
21 106 119
|
3eqtr4a |
⊢ ( ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) ∧ ¬ 𝑧 ∈ 𝑉 ) → ∪ 𝑥 ∈ ( { 𝑧 } ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ) |
121 |
100 120
|
pm2.61dan |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ∪ 𝑥 ∈ ( { 𝑧 } ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ) |
122 |
121
|
uneq2d |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ( ∪ 𝑥 ∈ ( ran 𝑦 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ∪ ∪ 𝑥 ∈ ( { 𝑧 } ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) = ( ∪ 𝑥 ∈ ( ran 𝑦 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ∪ ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ) ) |
123 |
88 122
|
eqtrd |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ∪ 𝑥 ∈ ( ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) = ( ∪ 𝑥 ∈ ( ran 𝑦 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ∪ ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ) ) |
124 |
76 123
|
eqeq12d |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ( ( ran ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ↔ ( ( ran ( 𝐹 ‘ 𝑦 ) ∩ 𝑉 ) ∪ ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ) = ( ∪ 𝑥 ∈ ( ran 𝑦 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ∪ ( ran ( 𝐹 ‘ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ) ) ) |
125 |
54 124
|
syl5ibr |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) ) → ( ( ran ( 𝐹 ‘ 𝑦 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑦 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) → ( ran ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ) |
126 |
125
|
expcom |
⊢ ( ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) → ( 𝐹 ∈ ran 𝑆 → ( ( ran ( 𝐹 ‘ 𝑦 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑦 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) → ( ran ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ) ) |
127 |
126
|
a2d |
⊢ ( ( 𝑦 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ∧ 𝑧 ∈ ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) ) → ( ( 𝐹 ∈ ran 𝑆 → ( ran ( 𝐹 ‘ 𝑦 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑦 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) → ( 𝐹 ∈ ran 𝑆 → ( ran ( 𝐹 ‘ ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran ( 𝑦 ++ 〈“ 𝑧 ”〉 ) ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ) ) |
128 |
24 32 40 48 53 127
|
wrdind |
⊢ ( 𝑋 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) → ( 𝐹 ∈ ran 𝑆 → ( ran ( 𝐹 ‘ 𝑋 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑋 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ) |
129 |
128
|
com12 |
⊢ ( 𝐹 ∈ ran 𝑆 → ( 𝑋 ∈ Word ( ( mCN ‘ 𝑇 ) ∪ 𝑉 ) → ( ran ( 𝐹 ‘ 𝑋 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑋 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ) |
130 |
10 129
|
sylbid |
⊢ ( 𝐹 ∈ ran 𝑆 → ( 𝑋 ∈ 𝑅 → ( ran ( 𝐹 ‘ 𝑋 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑋 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) ) |
131 |
130
|
imp |
⊢ ( ( 𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ 𝑅 ) → ( ran ( 𝐹 ‘ 𝑋 ) ∩ 𝑉 ) = ∪ 𝑥 ∈ ( ran 𝑋 ∩ 𝑉 ) ( ran ( 𝐹 ‘ 〈“ 𝑥 ”〉 ) ∩ 𝑉 ) ) |