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 8597 |
. 2
⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂)) |
9 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem4 8593 |
. . . . 5
⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
10 | | frn 6214 |
. . . . 5
⊢ (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → ran 𝑂 ⊆ 𝐴) |
11 | 9, 10 | syl 17 |
. . . 4
⊢ (𝜑 → ran 𝑂 ⊆ 𝐴) |
12 | | simprl 811 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏 ∈ 𝐴) |
13 | 6 | adantr 472 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 We 𝐴) |
14 | 7 | adantr 472 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑅 Se 𝐴) |
15 | 1, 2, 3, 4, 5, 13,
14 | ordtypelem8 8597 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂)) |
16 | | isof1o 6737 |
. . . . . . . . . . . . 13
⊢ (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) → 𝑂:dom 𝑂–1-1-onto→ran
𝑂) |
17 | | f1of 6299 |
. . . . . . . . . . . . 13
⊢ (𝑂:dom 𝑂–1-1-onto→ran
𝑂 → 𝑂:dom 𝑂⟶ran 𝑂) |
18 | 15, 16, 17 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂:dom 𝑂⟶ran 𝑂) |
19 | | f1of1 6298 |
. . . . . . . . . . . . . 14
⊢ (𝑂:dom 𝑂–1-1-onto→ran
𝑂 → 𝑂:dom 𝑂–1-1→ran 𝑂) |
20 | 15, 16, 19 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂:dom 𝑂–1-1→ran 𝑂) |
21 | | simpl 474 |
. . . . . . . . . . . . . . 15
⊢ ((𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂) → 𝑏 ∈ 𝐴) |
22 | | seex 5229 |
. . . . . . . . . . . . . . 15
⊢ ((𝑅 Se 𝐴 ∧ 𝑏 ∈ 𝐴) → {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ∈ V) |
23 | 7, 21, 22 | syl2an 495 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ∈ V) |
24 | 11 | adantr 472 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ⊆ 𝐴) |
25 | | rexnal 3133 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∃𝑚 ∈ dom
𝑂 ¬ (𝑂‘𝑚)𝑅𝑏 ↔ ¬ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏) |
26 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem7 8596 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑚 ∈ dom 𝑂) → ((𝑂‘𝑚)𝑅𝑏 ∨ 𝑏 ∈ ran 𝑂)) |
27 | 26 | ord 391 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑚 ∈ dom 𝑂) → (¬ (𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂)) |
28 | 27 | rexlimdva 3169 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (∃𝑚 ∈ dom 𝑂 ¬ (𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂)) |
29 | 25, 28 | syl5bir 233 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏 → 𝑏 ∈ ran 𝑂)) |
30 | 29 | con1d 139 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑏 ∈ ran 𝑂 → ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏)) |
31 | 30 | impr 650 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏) |
32 | | ffun 6209 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → Fun 𝑂) |
33 | 9, 32 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → Fun 𝑂) |
34 | | funfn 6079 |
. . . . . . . . . . . . . . . . . . 19
⊢ (Fun
𝑂 ↔ 𝑂 Fn dom 𝑂) |
35 | 33, 34 | sylib 208 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑂 Fn dom 𝑂) |
36 | 35 | adantr 472 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Fn dom 𝑂) |
37 | | breq1 4807 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 = (𝑂‘𝑚) → (𝑐𝑅𝑏 ↔ (𝑂‘𝑚)𝑅𝑏)) |
38 | 37 | ralrn 6526 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑂 Fn dom 𝑂 → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏)) |
39 | 36, 38 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → (∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏 ↔ ∀𝑚 ∈ dom 𝑂(𝑂‘𝑚)𝑅𝑏)) |
40 | 31, 39 | mpbird 247 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏) |
41 | | ssrab 3821 |
. . . . . . . . . . . . . . 15
⊢ (ran
𝑂 ⊆ {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏} ↔ (ran 𝑂 ⊆ 𝐴 ∧ ∀𝑐 ∈ ran 𝑂 𝑐𝑅𝑏)) |
42 | 24, 40, 41 | sylanbrc 701 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ⊆ {𝑐 ∈ 𝐴 ∣ 𝑐𝑅𝑏}) |
43 | 23, 42 | ssexd 4957 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 ∈ V) |
44 | | f1dmex 7302 |
. . . . . . . . . . . . 13
⊢ ((𝑂:dom 𝑂–1-1→ran 𝑂 ∧ ran 𝑂 ∈ V) → dom 𝑂 ∈ V) |
45 | 20, 43, 44 | syl2anc 696 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → dom 𝑂 ∈ V) |
46 | | fex 6654 |
. . . . . . . . . . . 12
⊢ ((𝑂:dom 𝑂⟶ran 𝑂 ∧ dom 𝑂 ∈ V) → 𝑂 ∈ V) |
47 | 18, 45, 46 | syl2anc 696 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 ∈ V) |
48 | 1, 2, 3, 4, 5, 13,
14, 47 | ordtypelem9 8598 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)) |
49 | | isof1o 6737 |
. . . . . . . . . 10
⊢ (𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴) → 𝑂:dom 𝑂–1-1-onto→𝐴) |
50 | | f1ofo 6306 |
. . . . . . . . . 10
⊢ (𝑂:dom 𝑂–1-1-onto→𝐴 → 𝑂:dom 𝑂–onto→𝐴) |
51 | | forn 6280 |
. . . . . . . . . 10
⊢ (𝑂:dom 𝑂–onto→𝐴 → ran 𝑂 = 𝐴) |
52 | 48, 49, 50, 51 | 4syl 19 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → ran 𝑂 = 𝐴) |
53 | 12, 52 | eleqtrrd 2842 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ ran 𝑂)) → 𝑏 ∈ ran 𝑂) |
54 | 53 | expr 644 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑏 ∈ ran 𝑂 → 𝑏 ∈ ran 𝑂)) |
55 | 54 | pm2.18d 124 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ ran 𝑂) |
56 | 55 | ex 449 |
. . . . 5
⊢ (𝜑 → (𝑏 ∈ 𝐴 → 𝑏 ∈ ran 𝑂)) |
57 | 56 | ssrdv 3750 |
. . . 4
⊢ (𝜑 → 𝐴 ⊆ ran 𝑂) |
58 | 11, 57 | eqssd 3761 |
. . 3
⊢ (𝜑 → ran 𝑂 = 𝐴) |
59 | | isoeq5 6735 |
. . 3
⊢ (ran
𝑂 = 𝐴 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))) |
60 | 58, 59 | syl 17 |
. 2
⊢ (𝜑 → (𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂) ↔ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))) |
61 | 8, 60 | mpbid 222 |
1
⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)) |