Proof of Theorem dfsmo2
Step | Hyp | Ref
| Expression |
1 | | df-smo 8165 |
. 2
⊢ (Smo
𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)))) |
2 | | ralcom 3164 |
. . . . . 6
⊢
(∀𝑦 ∈
dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))) |
3 | | impexp 451 |
. . . . . . . . 9
⊢ (((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ (𝑦 ∈ dom 𝐹 → (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)))) |
4 | | simpr 485 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥) |
5 | | ordtr1 6303 |
. . . . . . . . . . . . . . 15
⊢ (Ord dom
𝐹 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)) |
6 | 5 | 3impib 1115 |
. . . . . . . . . . . . . 14
⊢ ((Ord dom
𝐹 ∧ 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹) |
7 | 6 | 3com23 1125 |
. . . . . . . . . . . . 13
⊢ ((Ord dom
𝐹 ∧ 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝐹) |
8 | | simp3 1137 |
. . . . . . . . . . . . 13
⊢ ((Ord dom
𝐹 ∧ 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥) |
9 | 7, 8 | jca 512 |
. . . . . . . . . . . 12
⊢ ((Ord dom
𝐹 ∧ 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → (𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥)) |
10 | 9 | 3expia 1120 |
. . . . . . . . . . 11
⊢ ((Ord dom
𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝑦 ∈ 𝑥 → (𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥))) |
11 | 4, 10 | impbid2 225 |
. . . . . . . . . 10
⊢ ((Ord dom
𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) ↔ 𝑦 ∈ 𝑥)) |
12 | 11 | imbi1d 342 |
. . . . . . . . 9
⊢ ((Ord dom
𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)))) |
13 | 3, 12 | bitr3id 285 |
. . . . . . . 8
⊢ ((Ord dom
𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝑦 ∈ dom 𝐹 → (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))) ↔ (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)))) |
14 | 13 | ralbidv2 3106 |
. . . . . . 7
⊢ ((Ord dom
𝐹 ∧ 𝑥 ∈ dom 𝐹) → (∀𝑦 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))) |
15 | 14 | ralbidva 3107 |
. . . . . 6
⊢ (Ord dom
𝐹 → (∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))) |
16 | 2, 15 | bitrid 282 |
. . . . 5
⊢ (Ord dom
𝐹 → (∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))) |
17 | 16 | pm5.32i 575 |
. . . 4
⊢ ((Ord dom
𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))) ↔ (Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))) |
18 | 17 | anbi2i 623 |
. . 3
⊢ ((𝐹:dom 𝐹⟶On ∧ (Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)))) ↔ (𝐹:dom 𝐹⟶On ∧ (Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))) |
19 | | 3anass 1094 |
. . 3
⊢ ((𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))) ↔ (𝐹:dom 𝐹⟶On ∧ (Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))))) |
20 | | 3anass 1094 |
. . 3
⊢ ((𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ (𝐹:dom 𝐹⟶On ∧ (Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))) |
21 | 18, 19, 20 | 3bitr4i 303 |
. 2
⊢ ((𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))) ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))) |
22 | 1, 21 | bitri 274 |
1
⊢ (Smo
𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))) |