| Step | Hyp | Ref
| Expression |
| 1 | | fin1a2lem.aa |
. . . 4
⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) |
| 2 | 1 | fin1a2lem2 10441 |
. . 3
⊢ 𝑆:On–1-1→On |
| 3 | | fin1a2lem.b |
. . . . 5
⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o
·o 𝑥)) |
| 4 | 3 | fin1a2lem4 10443 |
. . . 4
⊢ 𝐸:ω–1-1→ω |
| 5 | | f1f 6804 |
. . . 4
⊢ (𝐸:ω–1-1→ω → 𝐸:ω⟶ω) |
| 6 | | frn 6743 |
. . . . 5
⊢ (𝐸:ω⟶ω →
ran 𝐸 ⊆
ω) |
| 7 | | omsson 7891 |
. . . . 5
⊢ ω
⊆ On |
| 8 | 6, 7 | sstrdi 3996 |
. . . 4
⊢ (𝐸:ω⟶ω →
ran 𝐸 ⊆
On) |
| 9 | 4, 5, 8 | mp2b 10 |
. . 3
⊢ ran 𝐸 ⊆ On |
| 10 | | f1ores 6862 |
. . 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 692 |
. 2
⊢ (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(𝑆 “ ran 𝐸) |
| 12 | 9 | sseli 3979 |
. . . . . . . . 9
⊢ (𝑏 ∈ ran 𝐸 → 𝑏 ∈ On) |
| 13 | 1 | fin1a2lem1 10440 |
. . . . . . . . 9
⊢ (𝑏 ∈ On → (𝑆‘𝑏) = suc 𝑏) |
| 14 | 12, 13 | syl 17 |
. . . . . . . 8
⊢ (𝑏 ∈ ran 𝐸 → (𝑆‘𝑏) = suc 𝑏) |
| 15 | 14 | eqeq1d 2739 |
. . . . . . 7
⊢ (𝑏 ∈ ran 𝐸 → ((𝑆‘𝑏) = 𝑎 ↔ suc 𝑏 = 𝑎)) |
| 16 | 15 | rexbiia 3092 |
. . . . . 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 3979 |
. . . . . . . . . . 11
⊢ (𝑏 ∈ ran 𝐸 → 𝑏 ∈ ω) |
| 19 | | peano2 7912 |
. . . . . . . . . . 11
⊢ (𝑏 ∈ ω → suc 𝑏 ∈
ω) |
| 20 | 18, 19 | syl 17 |
. . . . . . . . . 10
⊢ (𝑏 ∈ ran 𝐸 → suc 𝑏 ∈ ω) |
| 21 | 3 | fin1a2lem5 10444 |
. . . . . . . . . . . 12
⊢ (𝑏 ∈ ω → (𝑏 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸)) |
| 22 | 21 | biimpd 229 |
. . . . . . . . . . 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 511 |
. . . . . . . . 9
⊢ (𝑏 ∈ ran 𝐸 → (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸)) |
| 25 | | eleq1 2829 |
. . . . . . . . . 10
⊢ (suc
𝑏 = 𝑎 → (suc 𝑏 ∈ ω ↔ 𝑎 ∈ ω)) |
| 26 | | eleq1 2829 |
. . . . . . . . . . 11
⊢ (suc
𝑏 = 𝑎 → (suc 𝑏 ∈ ran 𝐸 ↔ 𝑎 ∈ ran 𝐸)) |
| 27 | 26 | notbid 318 |
. . . . . . . . . 10
⊢ (suc
𝑏 = 𝑎 → (¬ suc 𝑏 ∈ ran 𝐸 ↔ ¬ 𝑎 ∈ ran 𝐸)) |
| 28 | 25, 27 | anbi12d 632 |
. . . . . . . . 9
⊢ (suc
𝑏 = 𝑎 → ((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))) |
| 29 | 24, 28 | syl5ibcom 245 |
. . . . . . . 8
⊢ (𝑏 ∈ ran 𝐸 → (suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))) |
| 30 | 29 | rexlimiv 3148 |
. . . . . . 7
⊢
(∃𝑏 ∈ ran
𝐸 suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)) |
| 31 | | peano1 7910 |
. . . . . . . . . . . . . 14
⊢ ∅
∈ ω |
| 32 | 3 | fin1a2lem3 10442 |
. . . . . . . . . . . . . 14
⊢ (∅
∈ ω → (𝐸‘∅) = (2o
·o ∅)) |
| 33 | 31, 32 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ (𝐸‘∅) = (2o
·o ∅) |
| 34 | | 2on 8520 |
. . . . . . . . . . . . . 14
⊢
2o ∈ On |
| 35 | | om0 8555 |
. . . . . . . . . . . . . 14
⊢
(2o ∈ On → (2o ·o
∅) = ∅) |
| 36 | 34, 35 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(2o ·o ∅) = ∅ |
| 37 | 33, 36 | eqtri 2765 |
. . . . . . . . . . . 12
⊢ (𝐸‘∅) =
∅ |
| 38 | | f1fun 6806 |
. . . . . . . . . . . . . 14
⊢ (𝐸:ω–1-1→ω → Fun 𝐸) |
| 39 | 4, 38 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ Fun 𝐸 |
| 40 | | f1dm 6808 |
. . . . . . . . . . . . . . 15
⊢ (𝐸:ω–1-1→ω → dom 𝐸 = ω) |
| 41 | 4, 40 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ dom 𝐸 = ω |
| 42 | 31, 41 | eleqtrri 2840 |
. . . . . . . . . . . . 13
⊢ ∅
∈ dom 𝐸 |
| 43 | | fvelrn 7096 |
. . . . . . . . . . . . 13
⊢ ((Fun
𝐸 ∧ ∅ ∈ dom
𝐸) → (𝐸‘∅) ∈ ran 𝐸) |
| 44 | 39, 42, 43 | mp2an 692 |
. . . . . . . . . . . 12
⊢ (𝐸‘∅) ∈ ran 𝐸 |
| 45 | 37, 44 | eqeltrri 2838 |
. . . . . . . . . . 11
⊢ ∅
∈ ran 𝐸 |
| 46 | | eleq1 2829 |
. . . . . . . . . . 11
⊢ (𝑎 = ∅ → (𝑎 ∈ ran 𝐸 ↔ ∅ ∈ ran 𝐸)) |
| 47 | 45, 46 | mpbiri 258 |
. . . . . . . . . 10
⊢ (𝑎 = ∅ → 𝑎 ∈ ran 𝐸) |
| 48 | 47 | necon3bi 2967 |
. . . . . . . . 9
⊢ (¬
𝑎 ∈ ran 𝐸 → 𝑎 ≠ ∅) |
| 49 | | nnsuc 7905 |
. . . . . . . . 9
⊢ ((𝑎 ∈ ω ∧ 𝑎 ≠ ∅) →
∃𝑏 ∈ ω
𝑎 = suc 𝑏) |
| 50 | 48, 49 | sylan2 593 |
. . . . . . . 8
⊢ ((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ω 𝑎 = suc 𝑏) |
| 51 | | eleq1 2829 |
. . . . . . . . . . . . 13
⊢ (𝑎 = suc 𝑏 → (𝑎 ∈ ω ↔ suc 𝑏 ∈ ω)) |
| 52 | | eleq1 2829 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = suc 𝑏 → (𝑎 ∈ ran 𝐸 ↔ suc 𝑏 ∈ ran 𝐸)) |
| 53 | 52 | notbid 318 |
. . . . . . . . . . . . 13
⊢ (𝑎 = suc 𝑏 → (¬ 𝑎 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸)) |
| 54 | 51, 53 | anbi12d 632 |
. . . . . . . . . . . 12
⊢ (𝑎 = suc 𝑏 → ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ↔ (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸))) |
| 55 | 54 | anbi1d 631 |
. . . . . . . . . . 11
⊢ (𝑎 = suc 𝑏 → (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) ↔ ((suc 𝑏 ∈ ω ∧ ¬ suc
𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω))) |
| 56 | | simplr 769 |
. . . . . . . . . . . 12
⊢ (((suc
𝑏 ∈ ω ∧
¬ suc 𝑏 ∈ ran
𝐸) ∧ 𝑏 ∈ ω) → ¬ suc 𝑏 ∈ ran 𝐸) |
| 57 | 21 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((suc
𝑏 ∈ ω ∧
¬ suc 𝑏 ∈ ran
𝐸) ∧ 𝑏 ∈ ω) → (𝑏 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸)) |
| 58 | 56, 57 | mpbird 257 |
. . . . . . . . . . 11
⊢ (((suc
𝑏 ∈ ω ∧
¬ suc 𝑏 ∈ ran
𝐸) ∧ 𝑏 ∈ ω) → 𝑏 ∈ ran 𝐸) |
| 59 | 55, 58 | biimtrdi 253 |
. . . . . . . . . 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 454 |
. . . . . . . 8
⊢ (((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑏 ∈ ran 𝐸) |
| 62 | | simprr 773 |
. . . . . . . . 9
⊢ (((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑎 = suc 𝑏) |
| 63 | 62 | eqcomd 2743 |
. . . . . . . 8
⊢ (((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → suc 𝑏 = 𝑎) |
| 64 | 50, 61, 63 | reximssdv 3173 |
. . . . . . 7
⊢ ((𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎) |
| 65 | 30, 64 | impbii 209 |
. . . . . 6
⊢
(∃𝑏 ∈ ran
𝐸 suc 𝑏 = 𝑎 ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)) |
| 66 | 16, 65 | bitri 275 |
. . . . 5
⊢
(∃𝑏 ∈ ran
𝐸(𝑆‘𝑏) = 𝑎 ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)) |
| 67 | | f1fn 6805 |
. . . . . . 7
⊢ (𝑆:On–1-1→On → 𝑆 Fn On) |
| 68 | 2, 67 | ax-mp 5 |
. . . . . 6
⊢ 𝑆 Fn On |
| 69 | | fvelimab 6981 |
. . . . . 6
⊢ ((𝑆 Fn On ∧ ran 𝐸 ⊆ On) → (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆‘𝑏) = 𝑎)) |
| 70 | 68, 9, 69 | mp2an 692 |
. . . . 5
⊢ (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆‘𝑏) = 𝑎) |
| 71 | | eldif 3961 |
. . . . 5
⊢ (𝑎 ∈ (ω ∖ ran
𝐸) ↔ (𝑎 ∈ ω ∧ ¬
𝑎 ∈ ran 𝐸)) |
| 72 | 66, 70, 71 | 3bitr4i 303 |
. . . 4
⊢ (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ 𝑎 ∈ (ω ∖ ran 𝐸)) |
| 73 | 72 | eqriv 2734 |
. . 3
⊢ (𝑆 “ ran 𝐸) = (ω ∖ ran 𝐸) |
| 74 | | f1oeq3 6838 |
. . 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 230 |
1
⊢ (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(ω ∖ ran 𝐸) |