Step | Hyp | Ref
| Expression |
1 | | fin1a2lem.aa |
. . . 4
⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) |
2 | 1 | fin1a2lem2 9509 |
. . 3
⊢ 𝑆:On–1-1→On |
3 | | fin1a2lem.b |
. . . . 5
⊢ 𝐸 = (𝑥 ∈ ω ↦
(2𝑜 ·𝑜 𝑥)) |
4 | 3 | fin1a2lem4 9511 |
. . . 4
⊢ 𝐸:ω–1-1→ω |
5 | | f1f 6314 |
. . . 4
⊢ (𝐸:ω–1-1→ω → 𝐸:ω⟶ω) |
6 | | frn 6260 |
. . . . 5
⊢ (𝐸:ω⟶ω →
ran 𝐸 ⊆
ω) |
7 | | omsson 7301 |
. . . . 5
⊢ ω
⊆ On |
8 | 6, 7 | syl6ss 3808 |
. . . 4
⊢ (𝐸:ω⟶ω →
ran 𝐸 ⊆
On) |
9 | 4, 5, 8 | mp2b 10 |
. . 3
⊢ ran 𝐸 ⊆ On |
10 | | f1ores 6368 |
. . 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 684 |
. 2
⊢ (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(𝑆 “ ran 𝐸) |
12 | 9 | sseli 3792 |
. . . . . . . . 9
⊢ (𝑏 ∈ ran 𝐸 → 𝑏 ∈ On) |
13 | 1 | fin1a2lem1 9508 |
. . . . . . . . 9
⊢ (𝑏 ∈ On → (𝑆‘𝑏) = suc 𝑏) |
14 | 12, 13 | syl 17 |
. . . . . . . 8
⊢ (𝑏 ∈ ran 𝐸 → (𝑆‘𝑏) = suc 𝑏) |
15 | 14 | eqeq1d 2799 |
. . . . . . 7
⊢ (𝑏 ∈ ran 𝐸 → ((𝑆‘𝑏) = 𝑎 ↔ suc 𝑏 = 𝑎)) |
16 | 15 | rexbiia 3219 |
. . . . . 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 3792 |
. . . . . . . . . . 11
⊢ (𝑏 ∈ ran 𝐸 → 𝑏 ∈ ω) |
19 | | peano2 7318 |
. . . . . . . . . . 11
⊢ (𝑏 ∈ ω → suc 𝑏 ∈
ω) |
20 | 18, 19 | syl 17 |
. . . . . . . . . 10
⊢ (𝑏 ∈ ran 𝐸 → suc 𝑏 ∈ ω) |
21 | 3 | fin1a2lem5 9512 |
. . . . . . . . . . . 12
⊢ (𝑏 ∈ ω → (𝑏 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸)) |
22 | 21 | biimpd 221 |
. . . . . . . . . . 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 508 |
. . . . . . . . 9
⊢ (𝑏 ∈ ran 𝐸 → (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸)) |
25 | | eleq1 2864 |
. . . . . . . . . 10
⊢ (suc
𝑏 = 𝑎 → (suc 𝑏 ∈ ω ↔ 𝑎 ∈ ω)) |
26 | | eleq1 2864 |
. . . . . . . . . . 11
⊢ (suc
𝑏 = 𝑎 → (suc 𝑏 ∈ ran 𝐸 ↔ 𝑎 ∈ ran 𝐸)) |
27 | 26 | notbid 310 |
. . . . . . . . . 10
⊢ (suc
𝑏 = 𝑎 → (¬ suc 𝑏 ∈ ran 𝐸 ↔ ¬ 𝑎 ∈ ran 𝐸)) |
28 | 25, 27 | anbi12d 625 |
. . . . . . . . 9
⊢ (suc
𝑏 = 𝑎 → ((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))) |
29 | 24, 28 | syl5ibcom 237 |
. . . . . . . 8
⊢ (𝑏 ∈ ran 𝐸 → (suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))) |
30 | 29 | rexlimiv 3206 |
. . . . . . 7
⊢
(∃𝑏 ∈ ran
𝐸 suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)) |
31 | | peano1 7317 |
. . . . . . . . . . . . . 14
⊢ ∅
∈ ω |
32 | 3 | fin1a2lem3 9510 |
. . . . . . . . . . . . . 14
⊢ (∅
∈ ω → (𝐸‘∅) = (2𝑜
·𝑜 ∅)) |
33 | 31, 32 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ (𝐸‘∅) =
(2𝑜 ·𝑜 ∅) |
34 | | 2on 7806 |
. . . . . . . . . . . . . 14
⊢
2𝑜 ∈ On |
35 | | om0 7835 |
. . . . . . . . . . . . . 14
⊢
(2𝑜 ∈ On → (2𝑜
·𝑜 ∅) = ∅) |
36 | 34, 35 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(2𝑜 ·𝑜 ∅) =
∅ |
37 | 33, 36 | eqtri 2819 |
. . . . . . . . . . . 12
⊢ (𝐸‘∅) =
∅ |
38 | | f1fun 6316 |
. . . . . . . . . . . . . 14
⊢ (𝐸:ω–1-1→ω → Fun 𝐸) |
39 | 4, 38 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ Fun 𝐸 |
40 | | f1dm 6318 |
. . . . . . . . . . . . . . 15
⊢ (𝐸:ω–1-1→ω → dom 𝐸 = ω) |
41 | 4, 40 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ dom 𝐸 = ω |
42 | 31, 41 | eleqtrri 2875 |
. . . . . . . . . . . . 13
⊢ ∅
∈ dom 𝐸 |
43 | | fvelrn 6576 |
. . . . . . . . . . . . 13
⊢ ((Fun
𝐸 ∧ ∅ ∈ dom
𝐸) → (𝐸‘∅) ∈ ran 𝐸) |
44 | 39, 42, 43 | mp2an 684 |
. . . . . . . . . . . 12
⊢ (𝐸‘∅) ∈ ran 𝐸 |
45 | 37, 44 | eqeltrri 2873 |
. . . . . . . . . . 11
⊢ ∅
∈ ran 𝐸 |
46 | | eleq1 2864 |
. . . . . . . . . . 11
⊢ (𝑎 = ∅ → (𝑎 ∈ ran 𝐸 ↔ ∅ ∈ ran 𝐸)) |
47 | 45, 46 | mpbiri 250 |
. . . . . . . . . 10
⊢ (𝑎 = ∅ → 𝑎 ∈ ran 𝐸) |
48 | 47 | necon3bi 2995 |
. . . . . . . . 9
⊢ (¬
𝑎 ∈ ran 𝐸 → 𝑎 ≠ ∅) |
49 | | nnsuc 7314 |
. . . . . . . . 9
⊢ ((𝑎 ∈ ω ∧ 𝑎 ≠ ∅) →
∃𝑏 ∈ ω
𝑎 = suc 𝑏) |
50 | 48, 49 | sylan2 587 |
. . . . . . . 8
⊢ ((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ω 𝑎 = suc 𝑏) |
51 | | eleq1 2864 |
. . . . . . . . . . . . 13
⊢ (𝑎 = suc 𝑏 → (𝑎 ∈ ω ↔ suc 𝑏 ∈ ω)) |
52 | | eleq1 2864 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = suc 𝑏 → (𝑎 ∈ ran 𝐸 ↔ suc 𝑏 ∈ ran 𝐸)) |
53 | 52 | notbid 310 |
. . . . . . . . . . . . 13
⊢ (𝑎 = suc 𝑏 → (¬ 𝑎 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸)) |
54 | 51, 53 | anbi12d 625 |
. . . . . . . . . . . 12
⊢ (𝑎 = suc 𝑏 → ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ↔ (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸))) |
55 | 54 | anbi1d 624 |
. . . . . . . . . . 11
⊢ (𝑎 = suc 𝑏 → (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) ↔ ((suc 𝑏 ∈ ω ∧ ¬ suc
𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω))) |
56 | | simplr 786 |
. . . . . . . . . . . 12
⊢ (((suc
𝑏 ∈ ω ∧
¬ suc 𝑏 ∈ ran
𝐸) ∧ 𝑏 ∈ ω) → ¬ suc 𝑏 ∈ ran 𝐸) |
57 | 21 | adantl 474 |
. . . . . . . . . . . 12
⊢ (((suc
𝑏 ∈ ω ∧
¬ suc 𝑏 ∈ ran
𝐸) ∧ 𝑏 ∈ ω) → (𝑏 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸)) |
58 | 56, 57 | mpbird 249 |
. . . . . . . . . . 11
⊢ (((suc
𝑏 ∈ ω ∧
¬ suc 𝑏 ∈ ran
𝐸) ∧ 𝑏 ∈ ω) → 𝑏 ∈ ran 𝐸) |
59 | 55, 58 | syl6bi 245 |
. . . . . . . . . 10
⊢ (𝑎 = suc 𝑏 → (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → 𝑏 ∈ ran 𝐸)) |
60 | 59 | com12 32 |
. . . . . . . . 9
⊢ (((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → (𝑎 = suc 𝑏 → 𝑏 ∈ ran 𝐸)) |
61 | 60 | impr 447 |
. . . . . . . 8
⊢ (((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑏 ∈ ran 𝐸) |
62 | | simprr 790 |
. . . . . . . . 9
⊢ (((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑎 = suc 𝑏) |
63 | 62 | eqcomd 2803 |
. . . . . . . 8
⊢ (((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → suc 𝑏 = 𝑎) |
64 | 50, 61, 63 | reximssdv 3197 |
. . . . . . 7
⊢ ((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎) |
65 | 30, 64 | impbii 201 |
. . . . . 6
⊢
(∃𝑏 ∈ ran
𝐸 suc 𝑏 = 𝑎 ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)) |
66 | 16, 65 | bitri 267 |
. . . . 5
⊢
(∃𝑏 ∈ ran
𝐸(𝑆‘𝑏) = 𝑎 ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)) |
67 | | f1fn 6315 |
. . . . . . 7
⊢ (𝑆:On–1-1→On → 𝑆 Fn On) |
68 | 2, 67 | ax-mp 5 |
. . . . . 6
⊢ 𝑆 Fn On |
69 | | fvelimab 6476 |
. . . . . 6
⊢ ((𝑆 Fn On ∧ ran 𝐸 ⊆ On) → (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆‘𝑏) = 𝑎)) |
70 | 68, 9, 69 | mp2an 684 |
. . . . 5
⊢ (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆‘𝑏) = 𝑎) |
71 | | eldif 3777 |
. . . . 5
⊢ (𝑎 ∈ (ω ∖ ran
𝐸) ↔ (𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸)) |
72 | 66, 70, 71 | 3bitr4i 295 |
. . . 4
⊢ (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ 𝑎 ∈ (ω ∖ ran 𝐸)) |
73 | 72 | eqriv 2794 |
. . 3
⊢ (𝑆 “ ran 𝐸) = (ω ∖ ran 𝐸) |
74 | | f1oeq3 6345 |
. . 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))) |
75 | 73, 74 | 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)) |
76 | 11, 75 | mpbi 222 |
1
⊢ (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(ω ∖ ran 𝐸) |