| Step | Hyp | Ref
| Expression |
| 1 | | ordtypelem.1 |
. . 3
⊢ 𝐹 = recs(𝐺) |
| 2 | | ordtypelem.2 |
. . 3
⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} |
| 3 | | ordtypelem.3 |
. . 3
⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) |
| 4 | | ordtypelem.5 |
. . 3
⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} |
| 5 | | ordtypelem.6 |
. . 3
⊢ 𝑂 = OrdIso(𝑅, 𝐴) |
| 6 | | ordtypelem.7 |
. . 3
⊢ (𝜑 → 𝑅 We 𝐴) |
| 7 | | ordtypelem.8 |
. . 3
⊢ (𝜑 → 𝑅 Se 𝐴) |
| 8 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem8 9565 |
. 2
⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂)) |
| 9 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem4 9561 |
. . . . 5
⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
| 10 | 9 | frnd 6744 |
. . . 4
⊢ (𝜑 → ran 𝑂 ⊆ 𝐴) |
| 11 | | simprl 771 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏 ∈ 𝐴) |
| 12 | 6 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 We 𝐴) |
| 13 | 7 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 Se 𝐴) |
| 14 | 9 | ffund 6740 |
. . . . . . . . . . . 12
⊢ (𝜑 → Fun 𝑂) |
| 15 | 14 | funfnd 6597 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑂 Fn dom 𝑂) |
| 16 | 15 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Fn dom 𝑂) |
| 17 | 1, 2, 3, 4, 5, 12,
13 | ordtypelem8 9565 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂)) |
| 18 | | isof1o 7343 |
. . . . . . . . . . . 12
⊢ (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) → 𝑂:dom 𝑂–1-1-onto→ran
𝑂) |
| 19 | | f1of1 6847 |
. . . . . . . . . . . 12
⊢ (𝑂:dom 𝑂–1-1-onto→ran
𝑂 → 𝑂:dom 𝑂–1-1→ran 𝑂) |
| 20 | 17, 18, 19 | 3syl 18 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂:dom 𝑂–1-1→ran 𝑂) |
| 21 | | simpl 482 |
. . . . . . . . . . . . 13
⊢ ((𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂) → 𝑏 ∈ 𝐴) |
| 22 | | seex 5644 |
. . . . . . . . . . . . 13
⊢ ((𝑅 Se 𝐴 ∧ 𝑏 ∈ 𝐴) → {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ∈ V) |
| 23 | 7, 21, 22 | syl2an 596 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ∈ V) |
| 24 | 10 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ⊆ 𝐴) |
| 25 | | rexnal 3100 |
. . . . . . . . . . . . . . . . 17
⊢
(∃𝑚 ∈ dom
𝑂 ¬ (𝑂‘𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏) |
| 26 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem7 9564 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂‘𝑚)𝑅𝑏 ∨ 𝑏 ∈ ran 𝑂)) |
| 27 | 26 | ord 865 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂)) |
| 28 | 27 | rexlimdva 3155 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂)) |
| 29 | 25, 28 | biimtrrid 243 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂)) |
| 30 | 29 | con1d 145 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑏 ∈ ran 𝑂 → ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏)) |
| 31 | 30 | impr 454 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏) |
| 32 | | breq1 5146 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐 = (𝑂‘𝑚) → (𝑐𝑅𝑏 ↔ (𝑂‘𝑚)𝑅𝑏)) |
| 33 | 32 | ralrn 7108 |
. . . . . . . . . . . . . . 15
⊢ (𝑂 Fn dom 𝑂 → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏)) |
| 34 | 16, 33 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏)) |
| 35 | 31, 34 | mpbird 257 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏) |
| 36 | | ssrab 4073 |
. . . . . . . . . . . . 13
⊢ (ran
𝑂 ⊆ {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ↔ (ran 𝑂 ⊆ 𝐴 ∧ ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏)) |
| 37 | 24, 35, 36 | sylanbrc 583 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ⊆ {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏}) |
| 38 | 23, 37 | ssexd 5324 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ∈ V) |
| 39 | | f1dmex 7981 |
. . . . . . . . . . 11
⊢ ((𝑂:dom 𝑂–1-1→ran 𝑂 ∧ ran 𝑂 ∈ V) → dom 𝑂 ∈ V) |
| 40 | 20, 38, 39 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → dom 𝑂 ∈ V) |
| 41 | 16, 40 | fnexd 7238 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 ∈ V) |
| 42 | 1, 2, 3, 4, 5, 12,
13, 41 | ordtypelem9 9566 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)) |
| 43 | | isof1o 7343 |
. . . . . . . 8
⊢ (𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴) → 𝑂:dom 𝑂–1-1-onto→𝐴) |
| 44 | | f1ofo 6855 |
. . . . . . . 8
⊢ (𝑂:dom 𝑂–1-1-onto→𝐴 → 𝑂:dom 𝑂–onto→𝐴) |
| 45 | | forn 6823 |
. . . . . . . 8
⊢ (𝑂:dom 𝑂–onto→𝐴 → ran 𝑂 = 𝐴) |
| 46 | 42, 43, 44, 45 | 4syl 19 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 = 𝐴) |
| 47 | 11, 46 | eleqtrrd 2844 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏 ∈ ran 𝑂) |
| 48 | 47 | expr 456 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑏 ∈ ran 𝑂 → 𝑏 ∈ ran 𝑂)) |
| 49 | 48 | pm2.18d 127 |
. . . 4
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ ran 𝑂) |
| 50 | 10, 49 | eqelssd 4005 |
. . 3
⊢ (𝜑 → ran 𝑂 = 𝐴) |
| 51 | | isoeq5 7341 |
. . 3
⊢ (ran
𝑂 = 𝐴 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))) |
| 52 | 50, 51 | syl 17 |
. 2
⊢ (𝜑 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))) |
| 53 | 8, 52 | mpbid 232 |
1
⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)) |