| Step | Hyp | Ref
| Expression |
| 1 | | fr0g 8472 |
. . . 4
⊢ (𝑥 ∈ V → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) = 𝑥) |
| 2 | 1 | elv 3484 |
. . 3
⊢
((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾
ω)‘∅) = 𝑥 |
| 3 | | frfnom 8471 |
. . . 4
⊢
(rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω) Fn
ω |
| 4 | | peano1 7906 |
. . . 4
⊢ ∅
∈ ω |
| 5 | | fnfvelrn 7098 |
. . . 4
⊢
(((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω) Fn ω
∧ ∅ ∈ ω) → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) ∈ ran
(rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω)) |
| 6 | 3, 4, 5 | mp2an 692 |
. . 3
⊢
((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾
ω)‘∅) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) |
| 7 | 2, 6 | eqeltrri 2837 |
. 2
⊢ 𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) |
| 8 | | fvelrnb 6967 |
. . . . 5
⊢
((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω) Fn ω
→ (𝑧 ∈ ran
(rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω) ↔ ∃𝑓 ∈ ω ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧)) |
| 9 | 3, 8 | ax-mp 5 |
. . . 4
⊢ (𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ↔ ∃𝑓 ∈ ω ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧) |
| 10 | | fvex 6917 |
. . . . . . . . . 10
⊢
((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V |
| 11 | 10 | sucid 6464 |
. . . . . . . . 9
⊢
((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) |
| 12 | 10 | sucex 7822 |
. . . . . . . . . 10
⊢ suc
((rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V |
| 13 | | eqid 2736 |
. . . . . . . . . . 11
⊢
(rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω) = (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) |
| 14 | | suceq 6448 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑣 → suc 𝑧 = suc 𝑣) |
| 15 | | suceq 6448 |
. . . . . . . . . . 11
⊢ (𝑧 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) → suc 𝑧 = suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓)) |
| 16 | 13, 14, 15 | frsucmpt2 8476 |
. . . . . . . . . 10
⊢ ((𝑓 ∈ ω ∧ suc
((rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V) → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) = suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓)) |
| 17 | 12, 16 | mpan2 691 |
. . . . . . . . 9
⊢ (𝑓 ∈ ω →
((rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) = suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓)) |
| 18 | 11, 17 | eleqtrrid 2847 |
. . . . . . . 8
⊢ (𝑓 ∈ ω →
((rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓)) |
| 19 | | eleq1 2828 |
. . . . . . . 8
⊢
(((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ↔ 𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓))) |
| 20 | 18, 19 | imbitrid 244 |
. . . . . . 7
⊢
(((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → (𝑓 ∈ ω → 𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓))) |
| 21 | | peano2b 7900 |
. . . . . . . 8
⊢ (𝑓 ∈ ω ↔ suc 𝑓 ∈
ω) |
| 22 | | fnfvelrn 7098 |
. . . . . . . . 9
⊢
(((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω) Fn ω
∧ suc 𝑓 ∈ ω)
→ ((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω)‘suc
𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)) |
| 23 | 3, 22 | mpan 690 |
. . . . . . . 8
⊢ (suc
𝑓 ∈ ω →
((rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)) |
| 24 | 21, 23 | sylbi 217 |
. . . . . . 7
⊢ (𝑓 ∈ ω →
((rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)) |
| 25 | 20, 24 | jca2 513 |
. . . . . 6
⊢
(((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → (𝑓 ∈ ω → (𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∧ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))) |
| 26 | | fvex 6917 |
. . . . . . 7
⊢
((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω)‘suc
𝑓) ∈
V |
| 27 | | eleq2 2829 |
. . . . . . . 8
⊢ (𝑤 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓))) |
| 28 | | eleq1 2828 |
. . . . . . . 8
⊢ (𝑤 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) → (𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ↔ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))) |
| 29 | 27, 28 | anbi12d 632 |
. . . . . . 7
⊢ (𝑤 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)) ↔ (𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∧ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))) |
| 30 | 26, 29 | spcev 3605 |
. . . . . 6
⊢ ((𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∧ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))) |
| 31 | 25, 30 | syl6com 37 |
. . . . 5
⊢ (𝑓 ∈ ω →
(((rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))) |
| 32 | 31 | rexlimiv 3147 |
. . . 4
⊢
(∃𝑓 ∈
ω ((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))) |
| 33 | 9, 32 | sylbi 217 |
. . 3
⊢ (𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))) |
| 34 | 33 | ax-gen 1795 |
. 2
⊢
∀𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))) |
| 35 | | fndm 6669 |
. . . . . 6
⊢
((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω) Fn ω
→ dom (rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω) =
ω) |
| 36 | 3, 35 | ax-mp 5 |
. . . . 5
⊢ dom
(rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω) = ω |
| 37 | | id 22 |
. . . . 5
⊢ (ω
∈ 𝑉 → ω
∈ 𝑉) |
| 38 | 36, 37 | eqeltrid 2844 |
. . . 4
⊢ (ω
∈ 𝑉 → dom
(rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω) ∈ 𝑉) |
| 39 | | fnfun 6666 |
. . . . 5
⊢
((rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾ ω) Fn ω
→ Fun (rec((𝑣 ∈ V
↦ suc 𝑣), 𝑥) ↾
ω)) |
| 40 | 3, 39 | ax-mp 5 |
. . . 4
⊢ Fun
(rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω) |
| 41 | | funrnex 7974 |
. . . 4
⊢ (dom
(rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω) ∈ 𝑉 → (Fun (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈
V)) |
| 42 | 38, 40, 41 | mpisyl 21 |
. . 3
⊢ (ω
∈ 𝑉 → ran
(rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω) ∈ V) |
| 43 | | eleq2 2829 |
. . . . 5
⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))) |
| 44 | | eleq2 2829 |
. . . . . . 7
⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))) |
| 45 | | eleq2 2829 |
. . . . . . . . 9
⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑤 ∈ 𝑦 ↔ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))) |
| 46 | 45 | anbi2d 630 |
. . . . . . . 8
⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) ↔ (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))) |
| 47 | 46 | exbidv 1921 |
. . . . . . 7
⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))) |
| 48 | 44, 47 | imbi12d 344 |
. . . . . 6
⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ((𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)) ↔ (𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))) |
| 49 | 48 | albidv 1920 |
. . . . 5
⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)) ↔ ∀𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))) |
| 50 | 43, 49 | anbi12d 632 |
. . . 4
⊢ (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ((𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))) ↔ (𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∧ ∀𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))))) |
| 51 | 50 | spcegv 3596 |
. . 3
⊢ (ran
(rec((𝑣 ∈ V ↦
suc 𝑣), 𝑥) ↾ ω) ∈ V → ((𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∧ ∀𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))) → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))) |
| 52 | 42, 51 | syl 17 |
. 2
⊢ (ω
∈ 𝑉 → ((𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∧ ∀𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))) → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))) |
| 53 | 7, 34, 52 | mp2ani 698 |
1
⊢ (ω
∈ 𝑉 →
∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))) |