Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) |
2 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
3 |
|
elnn0uz |
⊢ ( 0 ∈ ℕ0 ↔ 0 ∈ ( ℤ≥ ‘ 0 ) ) |
4 |
2 3
|
mpbi |
⊢ 0 ∈ ( ℤ≥ ‘ 0 ) |
5 |
|
3nn0 |
⊢ 3 ∈ ℕ0 |
6 |
|
elnn0uz |
⊢ ( 3 ∈ ℕ0 ↔ 3 ∈ ( ℤ≥ ‘ 0 ) ) |
7 |
5 6
|
mpbi |
⊢ 3 ∈ ( ℤ≥ ‘ 0 ) |
8 |
|
uzss |
⊢ ( 3 ∈ ( ℤ≥ ‘ 0 ) → ( ℤ≥ ‘ 3 ) ⊆ ( ℤ≥ ‘ 0 ) ) |
9 |
7 8
|
ax-mp |
⊢ ( ℤ≥ ‘ 3 ) ⊆ ( ℤ≥ ‘ 0 ) |
10 |
9
|
sseli |
⊢ ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) → 𝐿 ∈ ( ℤ≥ ‘ 0 ) ) |
11 |
|
eluzfz |
⊢ ( ( 0 ∈ ( ℤ≥ ‘ 0 ) ∧ 𝐿 ∈ ( ℤ≥ ‘ 0 ) ) → 0 ∈ ( 0 ... 𝐿 ) ) |
12 |
4 10 11
|
sylancr |
⊢ ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) → 0 ∈ ( 0 ... 𝐿 ) ) |
13 |
12
|
adantr |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → 0 ∈ ( 0 ... 𝐿 ) ) |
14 |
1 13
|
ffvelrnd |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → ( 𝑃 ‘ 0 ) ∈ 𝑉 ) |
15 |
|
clel5 |
⊢ ( ( 𝑃 ‘ 0 ) ∈ 𝑉 ↔ ∃ 𝑎 ∈ 𝑉 ( 𝑃 ‘ 0 ) = 𝑎 ) |
16 |
14 15
|
sylib |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → ∃ 𝑎 ∈ 𝑉 ( 𝑃 ‘ 0 ) = 𝑎 ) |
17 |
|
1eluzge0 |
⊢ 1 ∈ ( ℤ≥ ‘ 0 ) |
18 |
|
1z |
⊢ 1 ∈ ℤ |
19 |
|
3z |
⊢ 3 ∈ ℤ |
20 |
|
1le3 |
⊢ 1 ≤ 3 |
21 |
|
eluz2 |
⊢ ( 3 ∈ ( ℤ≥ ‘ 1 ) ↔ ( 1 ∈ ℤ ∧ 3 ∈ ℤ ∧ 1 ≤ 3 ) ) |
22 |
18 19 20 21
|
mpbir3an |
⊢ 3 ∈ ( ℤ≥ ‘ 1 ) |
23 |
|
uzss |
⊢ ( 3 ∈ ( ℤ≥ ‘ 1 ) → ( ℤ≥ ‘ 3 ) ⊆ ( ℤ≥ ‘ 1 ) ) |
24 |
22 23
|
ax-mp |
⊢ ( ℤ≥ ‘ 3 ) ⊆ ( ℤ≥ ‘ 1 ) |
25 |
24
|
sseli |
⊢ ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) → 𝐿 ∈ ( ℤ≥ ‘ 1 ) ) |
26 |
|
eluzfz |
⊢ ( ( 1 ∈ ( ℤ≥ ‘ 0 ) ∧ 𝐿 ∈ ( ℤ≥ ‘ 1 ) ) → 1 ∈ ( 0 ... 𝐿 ) ) |
27 |
17 25 26
|
sylancr |
⊢ ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) → 1 ∈ ( 0 ... 𝐿 ) ) |
28 |
27
|
adantr |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → 1 ∈ ( 0 ... 𝐿 ) ) |
29 |
1 28
|
ffvelrnd |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → ( 𝑃 ‘ 1 ) ∈ 𝑉 ) |
30 |
|
clel5 |
⊢ ( ( 𝑃 ‘ 1 ) ∈ 𝑉 ↔ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) |
31 |
29 30
|
sylib |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) |
32 |
16 31
|
jca |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → ( ∃ 𝑎 ∈ 𝑉 ( 𝑃 ‘ 0 ) = 𝑎 ∧ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) ) |
33 |
|
2eluzge0 |
⊢ 2 ∈ ( ℤ≥ ‘ 0 ) |
34 |
|
uzuzle23 |
⊢ ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) → 𝐿 ∈ ( ℤ≥ ‘ 2 ) ) |
35 |
|
eluzfz |
⊢ ( ( 2 ∈ ( ℤ≥ ‘ 0 ) ∧ 𝐿 ∈ ( ℤ≥ ‘ 2 ) ) → 2 ∈ ( 0 ... 𝐿 ) ) |
36 |
33 34 35
|
sylancr |
⊢ ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) → 2 ∈ ( 0 ... 𝐿 ) ) |
37 |
36
|
adantr |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → 2 ∈ ( 0 ... 𝐿 ) ) |
38 |
1 37
|
ffvelrnd |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → ( 𝑃 ‘ 2 ) ∈ 𝑉 ) |
39 |
|
clel5 |
⊢ ( ( 𝑃 ‘ 2 ) ∈ 𝑉 ↔ ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ) |
40 |
38 39
|
sylib |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ) |
41 |
|
eluzfz |
⊢ ( ( 3 ∈ ( ℤ≥ ‘ 0 ) ∧ 𝐿 ∈ ( ℤ≥ ‘ 3 ) ) → 3 ∈ ( 0 ... 𝐿 ) ) |
42 |
7 41
|
mpan |
⊢ ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) → 3 ∈ ( 0 ... 𝐿 ) ) |
43 |
42
|
adantr |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → 3 ∈ ( 0 ... 𝐿 ) ) |
44 |
1 43
|
ffvelrnd |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → ( 𝑃 ‘ 3 ) ∈ 𝑉 ) |
45 |
|
clel5 |
⊢ ( ( 𝑃 ‘ 3 ) ∈ 𝑉 ↔ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) |
46 |
44 45
|
sylib |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) |
47 |
32 40 46
|
jca32 |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → ( ( ∃ 𝑎 ∈ 𝑉 ( 𝑃 ‘ 0 ) = 𝑎 ∧ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
48 |
|
r19.42v |
⊢ ( ∃ 𝑑 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ∃ 𝑑 ∈ 𝑉 ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
49 |
|
r19.42v |
⊢ ( ∃ 𝑑 ∈ 𝑉 ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ( 𝑃 ‘ 3 ) = 𝑑 ) ↔ ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) |
50 |
49
|
anbi2i |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ∃ 𝑑 ∈ 𝑉 ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
51 |
48 50
|
bitri |
⊢ ( ∃ 𝑑 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
52 |
51
|
rexbii |
⊢ ( ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ∃ 𝑐 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
53 |
52
|
2rexbii |
⊢ ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
54 |
|
r19.42v |
⊢ ( ∃ 𝑐 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ∃ 𝑐 ∈ 𝑉 ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
55 |
|
r19.41v |
⊢ ( ∃ 𝑐 ∈ 𝑉 ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ↔ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) |
56 |
55
|
anbi2i |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ∃ 𝑐 ∈ 𝑉 ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
57 |
54 56
|
bitri |
⊢ ( ∃ 𝑐 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
58 |
57
|
2rexbii |
⊢ ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
59 |
|
r19.41v |
⊢ ( ∃ 𝑏 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ( ∃ 𝑏 ∈ 𝑉 ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
60 |
|
r19.42v |
⊢ ( ∃ 𝑏 ∈ 𝑉 ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ↔ ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) ) |
61 |
60
|
anbi1i |
⊢ ( ( ∃ 𝑏 ∈ 𝑉 ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
62 |
59 61
|
bitri |
⊢ ( ∃ 𝑏 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
63 |
62
|
rexbii |
⊢ ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ∃ 𝑎 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
64 |
|
r19.41v |
⊢ ( ∃ 𝑎 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ( ∃ 𝑎 ∈ 𝑉 ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
65 |
|
r19.41v |
⊢ ( ∃ 𝑎 ∈ 𝑉 ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) ↔ ( ∃ 𝑎 ∈ 𝑉 ( 𝑃 ‘ 0 ) = 𝑎 ∧ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) ) |
66 |
65
|
anbi1i |
⊢ ( ( ∃ 𝑎 ∈ 𝑉 ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ( ( ∃ 𝑎 ∈ 𝑉 ( 𝑃 ‘ 0 ) = 𝑎 ∧ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
67 |
63 64 66
|
3bitri |
⊢ ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ( ( ∃ 𝑎 ∈ 𝑉 ( 𝑃 ‘ 0 ) = 𝑎 ∧ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
68 |
53 58 67
|
3bitri |
⊢ ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ( 𝑃 ‘ 3 ) = 𝑑 ) ) ↔ ( ( ∃ 𝑎 ∈ 𝑉 ( 𝑃 ‘ 0 ) = 𝑎 ∧ ∃ 𝑏 ∈ 𝑉 ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ∃ 𝑐 ∈ 𝑉 ( 𝑃 ‘ 2 ) = 𝑐 ∧ ∃ 𝑑 ∈ 𝑉 ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |
69 |
47 68
|
sylibr |
⊢ ( ( 𝐿 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑃 : ( 0 ... 𝐿 ) ⟶ 𝑉 ) → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( 𝑃 ‘ 0 ) = 𝑎 ∧ ( 𝑃 ‘ 1 ) = 𝑏 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝑐 ∧ ( 𝑃 ‘ 3 ) = 𝑑 ) ) ) |