Step |
Hyp |
Ref |
Expression |
1 |
|
3wlkd.p |
⊢ 𝑃 = 〈“ 𝐴 𝐵 𝐶 𝐷 ”〉 |
2 |
|
3wlkd.f |
⊢ 𝐹 = 〈“ 𝐽 𝐾 𝐿 ”〉 |
3 |
|
3wlkd.s |
⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) ) |
4 |
1 2 3
|
3wlkdlem3 |
⊢ ( 𝜑 → ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ) |
5 |
|
simpl |
⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( 𝑃 ‘ 0 ) = 𝐴 ) |
6 |
5
|
eleq1d |
⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( ( 𝑃 ‘ 0 ) ∈ 𝑉 ↔ 𝐴 ∈ 𝑉 ) ) |
7 |
|
simpr |
⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( 𝑃 ‘ 1 ) = 𝐵 ) |
8 |
7
|
eleq1d |
⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( ( 𝑃 ‘ 1 ) ∈ 𝑉 ↔ 𝐵 ∈ 𝑉 ) ) |
9 |
6 8
|
anbi12d |
⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( ( ( 𝑃 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 1 ) ∈ 𝑉 ) ↔ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) ) |
10 |
9
|
biimparc |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ) → ( ( 𝑃 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 1 ) ∈ 𝑉 ) ) |
11 |
|
c0ex |
⊢ 0 ∈ V |
12 |
|
1ex |
⊢ 1 ∈ V |
13 |
11 12
|
pm3.2i |
⊢ ( 0 ∈ V ∧ 1 ∈ V ) |
14 |
|
fveq2 |
⊢ ( 𝑘 = 0 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 0 ) ) |
15 |
14
|
eleq1d |
⊢ ( 𝑘 = 0 → ( ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ↔ ( 𝑃 ‘ 0 ) ∈ 𝑉 ) ) |
16 |
|
fveq2 |
⊢ ( 𝑘 = 1 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 1 ) ) |
17 |
16
|
eleq1d |
⊢ ( 𝑘 = 1 → ( ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ↔ ( 𝑃 ‘ 1 ) ∈ 𝑉 ) ) |
18 |
15 17
|
ralprg |
⊢ ( ( 0 ∈ V ∧ 1 ∈ V ) → ( ∀ 𝑘 ∈ { 0 , 1 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ↔ ( ( 𝑃 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 1 ) ∈ 𝑉 ) ) ) |
19 |
13 18
|
mp1i |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ) → ( ∀ 𝑘 ∈ { 0 , 1 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ↔ ( ( 𝑃 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 1 ) ∈ 𝑉 ) ) ) |
20 |
10 19
|
mpbird |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ) → ∀ 𝑘 ∈ { 0 , 1 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ) |
21 |
20
|
ex |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ∀ 𝑘 ∈ { 0 , 1 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ) ) |
22 |
|
simpl |
⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → ( 𝑃 ‘ 2 ) = 𝐶 ) |
23 |
22
|
eleq1d |
⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → ( ( 𝑃 ‘ 2 ) ∈ 𝑉 ↔ 𝐶 ∈ 𝑉 ) ) |
24 |
|
simpr |
⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → ( 𝑃 ‘ 3 ) = 𝐷 ) |
25 |
24
|
eleq1d |
⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → ( ( 𝑃 ‘ 3 ) ∈ 𝑉 ↔ 𝐷 ∈ 𝑉 ) ) |
26 |
23 25
|
anbi12d |
⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → ( ( ( 𝑃 ‘ 2 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 3 ) ∈ 𝑉 ) ↔ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) ) |
27 |
26
|
biimparc |
⊢ ( ( ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( 𝑃 ‘ 2 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 3 ) ∈ 𝑉 ) ) |
28 |
|
2ex |
⊢ 2 ∈ V |
29 |
|
3ex |
⊢ 3 ∈ V |
30 |
28 29
|
pm3.2i |
⊢ ( 2 ∈ V ∧ 3 ∈ V ) |
31 |
|
fveq2 |
⊢ ( 𝑘 = 2 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 2 ) ) |
32 |
31
|
eleq1d |
⊢ ( 𝑘 = 2 → ( ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ↔ ( 𝑃 ‘ 2 ) ∈ 𝑉 ) ) |
33 |
|
fveq2 |
⊢ ( 𝑘 = 3 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 3 ) ) |
34 |
33
|
eleq1d |
⊢ ( 𝑘 = 3 → ( ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ↔ ( 𝑃 ‘ 3 ) ∈ 𝑉 ) ) |
35 |
32 34
|
ralprg |
⊢ ( ( 2 ∈ V ∧ 3 ∈ V ) → ( ∀ 𝑘 ∈ { 2 , 3 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ↔ ( ( 𝑃 ‘ 2 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 3 ) ∈ 𝑉 ) ) ) |
36 |
30 35
|
mp1i |
⊢ ( ( ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ∀ 𝑘 ∈ { 2 , 3 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ↔ ( ( 𝑃 ‘ 2 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 3 ) ∈ 𝑉 ) ) ) |
37 |
27 36
|
mpbird |
⊢ ( ( ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ∀ 𝑘 ∈ { 2 , 3 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ) |
38 |
37
|
ex |
⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) → ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → ∀ 𝑘 ∈ { 2 , 3 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ) ) |
39 |
21 38
|
im2anan9 |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) → ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ∀ 𝑘 ∈ { 0 , 1 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ∧ ∀ 𝑘 ∈ { 2 , 3 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ) ) ) |
40 |
3 4 39
|
sylc |
⊢ ( 𝜑 → ( ∀ 𝑘 ∈ { 0 , 1 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ∧ ∀ 𝑘 ∈ { 2 , 3 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ) ) |
41 |
2
|
fveq2i |
⊢ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ 〈“ 𝐽 𝐾 𝐿 ”〉 ) |
42 |
|
s3len |
⊢ ( ♯ ‘ 〈“ 𝐽 𝐾 𝐿 ”〉 ) = 3 |
43 |
41 42
|
eqtri |
⊢ ( ♯ ‘ 𝐹 ) = 3 |
44 |
43
|
oveq2i |
⊢ ( 0 ... ( ♯ ‘ 𝐹 ) ) = ( 0 ... 3 ) |
45 |
|
fz0to3un2pr |
⊢ ( 0 ... 3 ) = ( { 0 , 1 } ∪ { 2 , 3 } ) |
46 |
44 45
|
eqtri |
⊢ ( 0 ... ( ♯ ‘ 𝐹 ) ) = ( { 0 , 1 } ∪ { 2 , 3 } ) |
47 |
46
|
raleqi |
⊢ ( ∀ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ↔ ∀ 𝑘 ∈ ( { 0 , 1 } ∪ { 2 , 3 } ) ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ) |
48 |
|
ralunb |
⊢ ( ∀ 𝑘 ∈ ( { 0 , 1 } ∪ { 2 , 3 } ) ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ↔ ( ∀ 𝑘 ∈ { 0 , 1 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ∧ ∀ 𝑘 ∈ { 2 , 3 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ) ) |
49 |
47 48
|
bitri |
⊢ ( ∀ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ↔ ( ∀ 𝑘 ∈ { 0 , 1 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ∧ ∀ 𝑘 ∈ { 2 , 3 } ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ) ) |
50 |
40 49
|
sylibr |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ) |