| Step | Hyp | Ref
| Expression |
| 1 | | simplr 769 |
. 2
⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω⟶𝐴) |
| 2 | | omsmolem 8695 |
. . . . . . . . 9
⊢ (𝑧 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑦 ∈ 𝑧 → (𝐹‘𝑦) ∈ (𝐹‘𝑧)))) |
| 3 | 2 | adantl 481 |
. . . . . . . 8
⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑦 ∈ 𝑧 → (𝐹‘𝑦) ∈ (𝐹‘𝑧)))) |
| 4 | 3 | imp 406 |
. . . . . . 7
⊢ (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥))) → (𝑦 ∈ 𝑧 → (𝐹‘𝑦) ∈ (𝐹‘𝑧))) |
| 5 | | omsmolem 8695 |
. . . . . . . . 9
⊢ (𝑦 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦)))) |
| 6 | 5 | adantr 480 |
. . . . . . . 8
⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦)))) |
| 7 | 6 | imp 406 |
. . . . . . 7
⊢ (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥))) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦))) |
| 8 | 4, 7 | orim12d 967 |
. . . . . 6
⊢ (((𝑦 ∈ ω ∧ 𝑧 ∈ ω) ∧ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥))) → ((𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦) → ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ∨ (𝐹‘𝑧) ∈ (𝐹‘𝑦)))) |
| 9 | 8 | ancoms 458 |
. . . . 5
⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦) → ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ∨ (𝐹‘𝑧) ∈ (𝐹‘𝑦)))) |
| 10 | 9 | con3d 152 |
. . . 4
⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (¬ ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ∨ (𝐹‘𝑧) ∈ (𝐹‘𝑦)) → ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))) |
| 11 | | ffvelcdm 7101 |
. . . . . . . . . . 11
⊢ ((𝐹:ω⟶𝐴 ∧ 𝑦 ∈ ω) → (𝐹‘𝑦) ∈ 𝐴) |
| 12 | | ssel 3977 |
. . . . . . . . . . 11
⊢ (𝐴 ⊆ On → ((𝐹‘𝑦) ∈ 𝐴 → (𝐹‘𝑦) ∈ On)) |
| 13 | 11, 12 | syl5 34 |
. . . . . . . . . 10
⊢ (𝐴 ⊆ On → ((𝐹:ω⟶𝐴 ∧ 𝑦 ∈ ω) → (𝐹‘𝑦) ∈ On)) |
| 14 | 13 | expdimp 452 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → (𝐹‘𝑦) ∈ On)) |
| 15 | | eloni 6394 |
. . . . . . . . 9
⊢ ((𝐹‘𝑦) ∈ On → Ord (𝐹‘𝑦)) |
| 16 | 14, 15 | syl6 35 |
. . . . . . . 8
⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑦 ∈ ω → Ord (𝐹‘𝑦))) |
| 17 | | ffvelcdm 7101 |
. . . . . . . . . . 11
⊢ ((𝐹:ω⟶𝐴 ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ 𝐴) |
| 18 | | ssel 3977 |
. . . . . . . . . . 11
⊢ (𝐴 ⊆ On → ((𝐹‘𝑧) ∈ 𝐴 → (𝐹‘𝑧) ∈ On)) |
| 19 | 17, 18 | syl5 34 |
. . . . . . . . . 10
⊢ (𝐴 ⊆ On → ((𝐹:ω⟶𝐴 ∧ 𝑧 ∈ ω) → (𝐹‘𝑧) ∈ On)) |
| 20 | 19 | expdimp 452 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ On)) |
| 21 | | eloni 6394 |
. . . . . . . . 9
⊢ ((𝐹‘𝑧) ∈ On → Ord (𝐹‘𝑧)) |
| 22 | 20, 21 | syl6 35 |
. . . . . . . 8
⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → (𝑧 ∈ ω → Ord (𝐹‘𝑧))) |
| 23 | 16, 22 | anim12d 609 |
. . . . . . 7
⊢ ((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) → ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (Ord (𝐹‘𝑦) ∧ Ord (𝐹‘𝑧)))) |
| 24 | 23 | imp 406 |
. . . . . 6
⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (Ord (𝐹‘𝑦) ∧ Ord (𝐹‘𝑧))) |
| 25 | | ordtri3 6420 |
. . . . . 6
⊢ ((Ord
(𝐹‘𝑦) ∧ Ord (𝐹‘𝑧)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ ¬ ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ∨ (𝐹‘𝑧) ∈ (𝐹‘𝑦)))) |
| 26 | 24, 25 | syl 17 |
. . . . 5
⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ ¬ ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ∨ (𝐹‘𝑧) ∈ (𝐹‘𝑦)))) |
| 27 | 26 | adantlr 715 |
. . . 4
⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ ¬ ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ∨ (𝐹‘𝑧) ∈ (𝐹‘𝑦)))) |
| 28 | | nnord 7895 |
. . . . . 6
⊢ (𝑦 ∈ ω → Ord 𝑦) |
| 29 | | nnord 7895 |
. . . . . 6
⊢ (𝑧 ∈ ω → Ord 𝑧) |
| 30 | | ordtri3 6420 |
. . . . . 6
⊢ ((Ord
𝑦 ∧ Ord 𝑧) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))) |
| 31 | 28, 29, 30 | syl2an 596 |
. . . . 5
⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))) |
| 32 | 31 | adantl 481 |
. . . 4
⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (𝑦 = 𝑧 ↔ ¬ (𝑦 ∈ 𝑧 ∨ 𝑧 ∈ 𝑦))) |
| 33 | 10, 27, 32 | 3imtr4d 294 |
. . 3
⊢ ((((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) ∧ (𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)) |
| 34 | 33 | ralrimivva 3202 |
. 2
⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)) |
| 35 | | dff13 7275 |
. 2
⊢ (𝐹:ω–1-1→𝐴 ↔ (𝐹:ω⟶𝐴 ∧ ∀𝑦 ∈ ω ∀𝑧 ∈ ω ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))) |
| 36 | 1, 34, 35 | sylanbrc 583 |
1
⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω–1-1→𝐴) |