Step | Hyp | Ref
| Expression |
1 | | fin1a2lem.aa |
. . . 4
⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) |
2 | 1 | fin1a2lem2 9223 |
. . 3
⊢ 𝑆:On–1-1→On |
3 | | fin1a2lem.b |
. . . . 5
⊢ 𝐸 = (𝑥 ∈ ω ↦
(2𝑜 ·𝑜 𝑥)) |
4 | 3 | fin1a2lem4 9225 |
. . . 4
⊢ 𝐸:ω–1-1→ω |
5 | | f1f 6101 |
. . . 4
⊢ (𝐸:ω–1-1→ω → 𝐸:ω⟶ω) |
6 | | frn 6053 |
. . . . 5
⊢ (𝐸:ω⟶ω →
ran 𝐸 ⊆
ω) |
7 | | omsson 7069 |
. . . . 5
⊢ ω
⊆ On |
8 | 6, 7 | syl6ss 3615 |
. . . 4
⊢ (𝐸:ω⟶ω →
ran 𝐸 ⊆
On) |
9 | 4, 5, 8 | mp2b 10 |
. . 3
⊢ ran 𝐸 ⊆ On |
10 | | f1ores 6151 |
. . 3
⊢ ((𝑆:On–1-1→On ∧ ran 𝐸 ⊆ On) → (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(𝑆 “ ran 𝐸)) |
11 | 2, 9, 10 | mp2an 708 |
. 2
⊢ (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(𝑆 “ ran 𝐸) |
12 | 9 | sseli 3599 |
. . . . . . . . 9
⊢ (𝑏 ∈ ran 𝐸 → 𝑏 ∈ On) |
13 | 1 | fin1a2lem1 9222 |
. . . . . . . . 9
⊢ (𝑏 ∈ On → (𝑆‘𝑏) = suc 𝑏) |
14 | 12, 13 | syl 17 |
. . . . . . . 8
⊢ (𝑏 ∈ ran 𝐸 → (𝑆‘𝑏) = suc 𝑏) |
15 | 14 | eqeq1d 2624 |
. . . . . . 7
⊢ (𝑏 ∈ ran 𝐸 → ((𝑆‘𝑏) = 𝑎 ↔ suc 𝑏 = 𝑎)) |
16 | 15 | rexbiia 3040 |
. . . . . 6
⊢
(∃𝑏 ∈ ran
𝐸(𝑆‘𝑏) = 𝑎 ↔ ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎) |
17 | 4, 5, 6 | mp2b 10 |
. . . . . . . . . . . 12
⊢ ran 𝐸 ⊆
ω |
18 | 17 | sseli 3599 |
. . . . . . . . . . 11
⊢ (𝑏 ∈ ran 𝐸 → 𝑏 ∈ ω) |
19 | | peano2 7086 |
. . . . . . . . . . 11
⊢ (𝑏 ∈ ω → suc 𝑏 ∈
ω) |
20 | 18, 19 | syl 17 |
. . . . . . . . . 10
⊢ (𝑏 ∈ ran 𝐸 → suc 𝑏 ∈ ω) |
21 | 3 | fin1a2lem5 9226 |
. . . . . . . . . . . 12
⊢ (𝑏 ∈ ω → (𝑏 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸)) |
22 | 21 | biimpd 219 |
. . . . . . . . . . 11
⊢ (𝑏 ∈ ω → (𝑏 ∈ ran 𝐸 → ¬ suc 𝑏 ∈ ran 𝐸)) |
23 | 18, 22 | mpcom 38 |
. . . . . . . . . 10
⊢ (𝑏 ∈ ran 𝐸 → ¬ suc 𝑏 ∈ ran 𝐸) |
24 | 20, 23 | jca 554 |
. . . . . . . . 9
⊢ (𝑏 ∈ ran 𝐸 → (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸)) |
25 | | eleq1 2689 |
. . . . . . . . . 10
⊢ (suc
𝑏 = 𝑎 → (suc 𝑏 ∈ ω ↔ 𝑎 ∈ ω)) |
26 | | eleq1 2689 |
. . . . . . . . . . 11
⊢ (suc
𝑏 = 𝑎 → (suc 𝑏 ∈ ran 𝐸 ↔ 𝑎 ∈ ran 𝐸)) |
27 | 26 | notbid 308 |
. . . . . . . . . 10
⊢ (suc
𝑏 = 𝑎 → (¬ suc 𝑏 ∈ ran 𝐸 ↔ ¬ 𝑎 ∈ ran 𝐸)) |
28 | 25, 27 | anbi12d 747 |
. . . . . . . . 9
⊢ (suc
𝑏 = 𝑎 → ((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))) |
29 | 24, 28 | syl5ibcom 235 |
. . . . . . . 8
⊢ (𝑏 ∈ ran 𝐸 → (suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))) |
30 | 29 | rexlimiv 3027 |
. . . . . . 7
⊢
(∃𝑏 ∈ ran
𝐸 suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)) |
31 | | peano1 7085 |
. . . . . . . . . . . . . 14
⊢ ∅
∈ ω |
32 | 3 | fin1a2lem3 9224 |
. . . . . . . . . . . . . 14
⊢ (∅
∈ ω → (𝐸‘∅) = (2𝑜
·𝑜 ∅)) |
33 | 31, 32 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ (𝐸‘∅) =
(2𝑜 ·𝑜 ∅) |
34 | | om0x 7599 |
. . . . . . . . . . . . 13
⊢
(2𝑜 ·𝑜 ∅) =
∅ |
35 | 33, 34 | eqtri 2644 |
. . . . . . . . . . . 12
⊢ (𝐸‘∅) =
∅ |
36 | | f1fun 6103 |
. . . . . . . . . . . . . 14
⊢ (𝐸:ω–1-1→ω → Fun 𝐸) |
37 | 4, 36 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ Fun 𝐸 |
38 | | f1dm 6105 |
. . . . . . . . . . . . . . 15
⊢ (𝐸:ω–1-1→ω → dom 𝐸 = ω) |
39 | 4, 38 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ dom 𝐸 = ω |
40 | 31, 39 | eleqtrri 2700 |
. . . . . . . . . . . . 13
⊢ ∅
∈ dom 𝐸 |
41 | | fvelrn 6352 |
. . . . . . . . . . . . 13
⊢ ((Fun
𝐸 ∧ ∅ ∈ dom
𝐸) → (𝐸‘∅) ∈ ran 𝐸) |
42 | 37, 40, 41 | mp2an 708 |
. . . . . . . . . . . 12
⊢ (𝐸‘∅) ∈ ran 𝐸 |
43 | 35, 42 | eqeltrri 2698 |
. . . . . . . . . . 11
⊢ ∅
∈ ran 𝐸 |
44 | | eleq1 2689 |
. . . . . . . . . . 11
⊢ (𝑎 = ∅ → (𝑎 ∈ ran 𝐸 ↔ ∅ ∈ ran 𝐸)) |
45 | 43, 44 | mpbiri 248 |
. . . . . . . . . 10
⊢ (𝑎 = ∅ → 𝑎 ∈ ran 𝐸) |
46 | 45 | necon3bi 2820 |
. . . . . . . . 9
⊢ (¬
𝑎 ∈ ran 𝐸 → 𝑎 ≠ ∅) |
47 | | nnsuc 7082 |
. . . . . . . . 9
⊢ ((𝑎 ∈ ω ∧ 𝑎 ≠ ∅) →
∃𝑏 ∈ ω
𝑎 = suc 𝑏) |
48 | 46, 47 | sylan2 491 |
. . . . . . . 8
⊢ ((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ω 𝑎 = suc 𝑏) |
49 | | eleq1 2689 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = suc 𝑏 → (𝑎 ∈ ω ↔ suc 𝑏 ∈ ω)) |
50 | | eleq1 2689 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = suc 𝑏 → (𝑎 ∈ ran 𝐸 ↔ suc 𝑏 ∈ ran 𝐸)) |
51 | 50 | notbid 308 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = suc 𝑏 → (¬ 𝑎 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸)) |
52 | 49, 51 | anbi12d 747 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = suc 𝑏 → ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ↔ (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸))) |
53 | 52 | anbi1d 741 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = suc 𝑏 → (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) ↔ ((suc 𝑏 ∈ ω ∧ ¬ suc
𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω))) |
54 | | simplr 792 |
. . . . . . . . . . . . . . 15
⊢ (((suc
𝑏 ∈ ω ∧
¬ suc 𝑏 ∈ ran
𝐸) ∧ 𝑏 ∈ ω) → ¬ suc 𝑏 ∈ ran 𝐸) |
55 | 21 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ (((suc
𝑏 ∈ ω ∧
¬ suc 𝑏 ∈ ran
𝐸) ∧ 𝑏 ∈ ω) → (𝑏 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸)) |
56 | 54, 55 | mpbird 247 |
. . . . . . . . . . . . . 14
⊢ (((suc
𝑏 ∈ ω ∧
¬ suc 𝑏 ∈ ran
𝐸) ∧ 𝑏 ∈ ω) → 𝑏 ∈ ran 𝐸) |
57 | 53, 56 | syl6bi 243 |
. . . . . . . . . . . . 13
⊢ (𝑎 = suc 𝑏 → (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → 𝑏 ∈ ran 𝐸)) |
58 | 57 | com12 32 |
. . . . . . . . . . . 12
⊢ (((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → (𝑎 = suc 𝑏 → 𝑏 ∈ ran 𝐸)) |
59 | 58 | impr 649 |
. . . . . . . . . . 11
⊢ (((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑏 ∈ ran 𝐸) |
60 | | simprr 796 |
. . . . . . . . . . . 12
⊢ (((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑎 = suc 𝑏) |
61 | 60 | eqcomd 2628 |
. . . . . . . . . . 11
⊢ (((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → suc 𝑏 = 𝑎) |
62 | 59, 61 | jca 554 |
. . . . . . . . . 10
⊢ (((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → (𝑏 ∈ ran 𝐸 ∧ suc 𝑏 = 𝑎)) |
63 | 62 | ex 450 |
. . . . . . . . 9
⊢ ((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) → ((𝑏 ∈ ω ∧ 𝑎 = suc 𝑏) → (𝑏 ∈ ran 𝐸 ∧ suc 𝑏 = 𝑎))) |
64 | 63 | reximdv2 3014 |
. . . . . . . 8
⊢ ((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) → (∃𝑏 ∈ ω 𝑎 = suc 𝑏 → ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎)) |
65 | 48, 64 | mpd 15 |
. . . . . . 7
⊢ ((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎) |
66 | 30, 65 | impbii 199 |
. . . . . 6
⊢
(∃𝑏 ∈ ran
𝐸 suc 𝑏 = 𝑎 ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)) |
67 | 16, 66 | bitri 264 |
. . . . 5
⊢
(∃𝑏 ∈ ran
𝐸(𝑆‘𝑏) = 𝑎 ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)) |
68 | | f1fn 6102 |
. . . . . . 7
⊢ (𝑆:On–1-1→On → 𝑆 Fn On) |
69 | 2, 68 | ax-mp 5 |
. . . . . 6
⊢ 𝑆 Fn On |
70 | | fvelimab 6253 |
. . . . . 6
⊢ ((𝑆 Fn On ∧ ran 𝐸 ⊆ On) → (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆‘𝑏) = 𝑎)) |
71 | 69, 9, 70 | mp2an 708 |
. . . . 5
⊢ (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆‘𝑏) = 𝑎) |
72 | | eldif 3584 |
. . . . 5
⊢ (𝑎 ∈ (ω ∖ ran
𝐸) ↔ (𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸)) |
73 | 67, 71, 72 | 3bitr4i 292 |
. . . 4
⊢ (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ 𝑎 ∈ (ω ∖ ran 𝐸)) |
74 | 73 | eqriv 2619 |
. . 3
⊢ (𝑆 “ ran 𝐸) = (ω ∖ ran 𝐸) |
75 | | f1oeq3 6129 |
. . 3
⊢ ((𝑆 “ ran 𝐸) = (ω ∖ ran 𝐸) → ((𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(𝑆 “ ran 𝐸) ↔ (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(ω ∖ ran 𝐸))) |
76 | 74, 75 | ax-mp 5 |
. 2
⊢ ((𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(𝑆 “ ran 𝐸) ↔ (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(ω ∖ ran 𝐸)) |
77 | 11, 76 | mpbi 220 |
1
⊢ (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(ω ∖ ran 𝐸) |