| Step | Hyp | Ref
| Expression |
| 1 | | epweon 7795 |
. . . . . 6
⊢ E We
On |
| 2 | | wess 5671 |
. . . . . 6
⊢ (𝐴 ⊆ On → ( E We On
→ E We 𝐴)) |
| 3 | 1, 2 | mpi 20 |
. . . . 5
⊢ (𝐴 ⊆ On → E We 𝐴) |
| 4 | | epse 5667 |
. . . . 5
⊢ E Se
𝐴 |
| 5 | | oismo.1 |
. . . . . 6
⊢ 𝐹 = OrdIso( E , 𝐴) |
| 6 | 5 | oiiso2 9571 |
. . . . 5
⊢ (( E We
𝐴 ∧ E Se 𝐴) → 𝐹 Isom E , E (dom 𝐹, ran 𝐹)) |
| 7 | 3, 4, 6 | sylancl 586 |
. . . 4
⊢ (𝐴 ⊆ On → 𝐹 Isom E , E (dom 𝐹, ran 𝐹)) |
| 8 | 5 | oicl 9569 |
. . . . 5
⊢ Ord dom
𝐹 |
| 9 | 5 | oif 9570 |
. . . . . . 7
⊢ 𝐹:dom 𝐹⟶𝐴 |
| 10 | | frn 6743 |
. . . . . . 7
⊢ (𝐹:dom 𝐹⟶𝐴 → ran 𝐹 ⊆ 𝐴) |
| 11 | 9, 10 | ax-mp 5 |
. . . . . 6
⊢ ran 𝐹 ⊆ 𝐴 |
| 12 | | id 22 |
. . . . . 6
⊢ (𝐴 ⊆ On → 𝐴 ⊆ On) |
| 13 | 11, 12 | sstrid 3995 |
. . . . 5
⊢ (𝐴 ⊆ On → ran 𝐹 ⊆ On) |
| 14 | | smoiso2 8409 |
. . . . 5
⊢ ((Ord dom
𝐹 ∧ ran 𝐹 ⊆ On) → ((𝐹:dom 𝐹–onto→ran 𝐹 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (dom 𝐹, ran 𝐹))) |
| 15 | 8, 13, 14 | sylancr 587 |
. . . 4
⊢ (𝐴 ⊆ On → ((𝐹:dom 𝐹–onto→ran 𝐹 ∧ Smo 𝐹) ↔ 𝐹 Isom E , E (dom 𝐹, ran 𝐹))) |
| 16 | 7, 15 | mpbird 257 |
. . 3
⊢ (𝐴 ⊆ On → (𝐹:dom 𝐹–onto→ran 𝐹 ∧ Smo 𝐹)) |
| 17 | 16 | simprd 495 |
. 2
⊢ (𝐴 ⊆ On → Smo 𝐹) |
| 18 | 11 | a1i 11 |
. . 3
⊢ (𝐴 ⊆ On → ran 𝐹 ⊆ 𝐴) |
| 19 | | simprl 771 |
. . . . . 6
⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥 ∈ 𝐴) |
| 20 | 3 | adantr 480 |
. . . . . . . 8
⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → E We 𝐴) |
| 21 | 4 | a1i 11 |
. . . . . . . 8
⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → E Se 𝐴) |
| 22 | | ffn 6736 |
. . . . . . . . . . 11
⊢ (𝐹:dom 𝐹⟶𝐴 → 𝐹 Fn dom 𝐹) |
| 23 | 9, 22 | mp1i 13 |
. . . . . . . . . 10
⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 Fn dom 𝐹) |
| 24 | | simplrr 778 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ¬ 𝑥 ∈ ran 𝐹) |
| 25 | 3 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → E We 𝐴) |
| 26 | 4 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → E Se 𝐴) |
| 27 | | simplrl 777 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥 ∈ 𝐴) |
| 28 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹) |
| 29 | 5 | oiiniseg 9573 |
. . . . . . . . . . . . . . 15
⊢ ((( E We
𝐴 ∧ E Se 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝐹)) → ((𝐹‘𝑦) E 𝑥 ∨ 𝑥 ∈ ran 𝐹)) |
| 30 | 25, 26, 27, 28, 29 | syl22anc 839 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑦) E 𝑥 ∨ 𝑥 ∈ ran 𝐹)) |
| 31 | 30 | ord 865 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (¬ (𝐹‘𝑦) E 𝑥 → 𝑥 ∈ ran 𝐹)) |
| 32 | 24, 31 | mt3d 148 |
. . . . . . . . . . . 12
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) E 𝑥) |
| 33 | | epel 5587 |
. . . . . . . . . . . 12
⊢ ((𝐹‘𝑦) E 𝑥 ↔ (𝐹‘𝑦) ∈ 𝑥) |
| 34 | 32, 33 | sylib 218 |
. . . . . . . . . . 11
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ 𝑥) |
| 35 | 34 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → ∀𝑦 ∈ dom 𝐹(𝐹‘𝑦) ∈ 𝑥) |
| 36 | | ffnfv 7139 |
. . . . . . . . . 10
⊢ (𝐹:dom 𝐹⟶𝑥 ↔ (𝐹 Fn dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹(𝐹‘𝑦) ∈ 𝑥)) |
| 37 | 23, 35, 36 | sylanbrc 583 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹:dom 𝐹⟶𝑥) |
| 38 | 9, 22 | mp1i 13 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝐹 Fn dom 𝐹) |
| 39 | 17 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → Smo 𝐹) |
| 40 | | smogt 8407 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 Fn dom 𝐹 ∧ Smo 𝐹 ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹‘𝑦)) |
| 41 | 38, 39, 28, 40 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ⊆ (𝐹‘𝑦)) |
| 42 | | ordelon 6408 |
. . . . . . . . . . . . . . 15
⊢ ((Ord dom
𝐹 ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ On) |
| 43 | 8, 28, 42 | sylancr 587 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ On) |
| 44 | | simpll 767 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝐴 ⊆ On) |
| 45 | 44, 27 | sseldd 3984 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑥 ∈ On) |
| 46 | | ontr2 6431 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ⊆ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ∈ 𝑥) → 𝑦 ∈ 𝑥)) |
| 47 | 43, 45, 46 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → ((𝑦 ⊆ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ∈ 𝑥) → 𝑦 ∈ 𝑥)) |
| 48 | 41, 34, 47 | mp2and 699 |
. . . . . . . . . . . 12
⊢ (((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) ∧ 𝑦 ∈ dom 𝐹) → 𝑦 ∈ 𝑥) |
| 49 | 48 | ex 412 |
. . . . . . . . . . 11
⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → (𝑦 ∈ dom 𝐹 → 𝑦 ∈ 𝑥)) |
| 50 | 49 | ssrdv 3989 |
. . . . . . . . . 10
⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹 ⊆ 𝑥) |
| 51 | 19, 50 | ssexd 5324 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → dom 𝐹 ∈ V) |
| 52 | | fex2 7958 |
. . . . . . . . 9
⊢ ((𝐹:dom 𝐹⟶𝑥 ∧ dom 𝐹 ∈ V ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ V) |
| 53 | 37, 51, 19, 52 | syl3anc 1373 |
. . . . . . . 8
⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 ∈ V) |
| 54 | 5 | ordtype2 9574 |
. . . . . . . 8
⊢ (( E We
𝐴 ∧ E Se 𝐴 ∧ 𝐹 ∈ V) → 𝐹 Isom E , E (dom 𝐹, 𝐴)) |
| 55 | 20, 21, 53, 54 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝐹 Isom E , E (dom 𝐹, 𝐴)) |
| 56 | | isof1o 7343 |
. . . . . . 7
⊢ (𝐹 Isom E , E (dom 𝐹, 𝐴) → 𝐹:dom 𝐹–1-1-onto→𝐴) |
| 57 | | f1ofo 6855 |
. . . . . . 7
⊢ (𝐹:dom 𝐹–1-1-onto→𝐴 → 𝐹:dom 𝐹–onto→𝐴) |
| 58 | | forn 6823 |
. . . . . . 7
⊢ (𝐹:dom 𝐹–onto→𝐴 → ran 𝐹 = 𝐴) |
| 59 | 55, 56, 57, 58 | 4syl 19 |
. . . . . 6
⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → ran 𝐹 = 𝐴) |
| 60 | 19, 59 | eleqtrrd 2844 |
. . . . 5
⊢ ((𝐴 ⊆ On ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran 𝐹)) → 𝑥 ∈ ran 𝐹) |
| 61 | 60 | expr 456 |
. . . 4
⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ ran 𝐹 → 𝑥 ∈ ran 𝐹)) |
| 62 | 61 | pm2.18d 127 |
. . 3
⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ran 𝐹) |
| 63 | 18, 62 | eqelssd 4005 |
. 2
⊢ (𝐴 ⊆ On → ran 𝐹 = 𝐴) |
| 64 | 17, 63 | jca 511 |
1
⊢ (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴)) |