Step |
Hyp |
Ref |
Expression |
1 |
|
ordtypelem.1 |
⊢ 𝐹 = recs ( 𝐺 ) |
2 |
|
ordtypelem.2 |
⊢ 𝐶 = { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } |
3 |
|
ordtypelem.3 |
⊢ 𝐺 = ( ℎ ∈ V ↦ ( ℩ 𝑣 ∈ 𝐶 ∀ 𝑢 ∈ 𝐶 ¬ 𝑢 𝑅 𝑣 ) ) |
4 |
|
ordtypelem.5 |
⊢ 𝑇 = { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 } |
5 |
|
ordtypelem.6 |
⊢ 𝑂 = OrdIso ( 𝑅 , 𝐴 ) |
6 |
|
ordtypelem.7 |
⊢ ( 𝜑 → 𝑅 We 𝐴 ) |
7 |
|
ordtypelem.8 |
⊢ ( 𝜑 → 𝑅 Se 𝐴 ) |
8 |
|
ordtypelem9.1 |
⊢ ( 𝜑 → 𝑂 ∈ 𝑉 ) |
9 |
1 2 3 4 5 6 7
|
ordtypelem8 |
⊢ ( 𝜑 → 𝑂 Isom E , 𝑅 ( dom 𝑂 , ran 𝑂 ) ) |
10 |
1 2 3 4 5 6 7
|
ordtypelem4 |
⊢ ( 𝜑 → 𝑂 : ( 𝑇 ∩ dom 𝐹 ) ⟶ 𝐴 ) |
11 |
10
|
frnd |
⊢ ( 𝜑 → ran 𝑂 ⊆ 𝐴 ) |
12 |
1 2 3 4 5 6 7
|
ordtypelem2 |
⊢ ( 𝜑 → Ord 𝑇 ) |
13 |
|
ordirr |
⊢ ( Ord 𝑇 → ¬ 𝑇 ∈ 𝑇 ) |
14 |
12 13
|
syl |
⊢ ( 𝜑 → ¬ 𝑇 ∈ 𝑇 ) |
15 |
1
|
tfr1a |
⊢ ( Fun 𝐹 ∧ Lim dom 𝐹 ) |
16 |
15
|
simpri |
⊢ Lim dom 𝐹 |
17 |
|
limord |
⊢ ( Lim dom 𝐹 → Ord dom 𝐹 ) |
18 |
16 17
|
ax-mp |
⊢ Ord dom 𝐹 |
19 |
1 2 3 4 5 6 7
|
ordtypelem1 |
⊢ ( 𝜑 → 𝑂 = ( 𝐹 ↾ 𝑇 ) ) |
20 |
8
|
elexd |
⊢ ( 𝜑 → 𝑂 ∈ V ) |
21 |
19 20
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝑇 ) ∈ V ) |
22 |
1
|
tfr2b |
⊢ ( Ord 𝑇 → ( 𝑇 ∈ dom 𝐹 ↔ ( 𝐹 ↾ 𝑇 ) ∈ V ) ) |
23 |
12 22
|
syl |
⊢ ( 𝜑 → ( 𝑇 ∈ dom 𝐹 ↔ ( 𝐹 ↾ 𝑇 ) ∈ V ) ) |
24 |
21 23
|
mpbird |
⊢ ( 𝜑 → 𝑇 ∈ dom 𝐹 ) |
25 |
|
ordelon |
⊢ ( ( Ord dom 𝐹 ∧ 𝑇 ∈ dom 𝐹 ) → 𝑇 ∈ On ) |
26 |
18 24 25
|
sylancr |
⊢ ( 𝜑 → 𝑇 ∈ On ) |
27 |
|
imaeq2 |
⊢ ( 𝑎 = 𝑇 → ( 𝐹 “ 𝑎 ) = ( 𝐹 “ 𝑇 ) ) |
28 |
27
|
raleqdv |
⊢ ( 𝑎 = 𝑇 → ( ∀ 𝑐 ∈ ( 𝐹 “ 𝑎 ) 𝑐 𝑅 𝑏 ↔ ∀ 𝑐 ∈ ( 𝐹 “ 𝑇 ) 𝑐 𝑅 𝑏 ) ) |
29 |
28
|
rexbidv |
⊢ ( 𝑎 = 𝑇 → ( ∃ 𝑏 ∈ 𝐴 ∀ 𝑐 ∈ ( 𝐹 “ 𝑎 ) 𝑐 𝑅 𝑏 ↔ ∃ 𝑏 ∈ 𝐴 ∀ 𝑐 ∈ ( 𝐹 “ 𝑇 ) 𝑐 𝑅 𝑏 ) ) |
30 |
|
breq1 |
⊢ ( 𝑧 = 𝑐 → ( 𝑧 𝑅 𝑡 ↔ 𝑐 𝑅 𝑡 ) ) |
31 |
30
|
cbvralvw |
⊢ ( ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 ↔ ∀ 𝑐 ∈ ( 𝐹 “ 𝑥 ) 𝑐 𝑅 𝑡 ) |
32 |
|
breq2 |
⊢ ( 𝑡 = 𝑏 → ( 𝑐 𝑅 𝑡 ↔ 𝑐 𝑅 𝑏 ) ) |
33 |
32
|
ralbidv |
⊢ ( 𝑡 = 𝑏 → ( ∀ 𝑐 ∈ ( 𝐹 “ 𝑥 ) 𝑐 𝑅 𝑡 ↔ ∀ 𝑐 ∈ ( 𝐹 “ 𝑥 ) 𝑐 𝑅 𝑏 ) ) |
34 |
31 33
|
syl5bb |
⊢ ( 𝑡 = 𝑏 → ( ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 ↔ ∀ 𝑐 ∈ ( 𝐹 “ 𝑥 ) 𝑐 𝑅 𝑏 ) ) |
35 |
34
|
cbvrexvw |
⊢ ( ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 ↔ ∃ 𝑏 ∈ 𝐴 ∀ 𝑐 ∈ ( 𝐹 “ 𝑥 ) 𝑐 𝑅 𝑏 ) |
36 |
|
imaeq2 |
⊢ ( 𝑥 = 𝑎 → ( 𝐹 “ 𝑥 ) = ( 𝐹 “ 𝑎 ) ) |
37 |
36
|
raleqdv |
⊢ ( 𝑥 = 𝑎 → ( ∀ 𝑐 ∈ ( 𝐹 “ 𝑥 ) 𝑐 𝑅 𝑏 ↔ ∀ 𝑐 ∈ ( 𝐹 “ 𝑎 ) 𝑐 𝑅 𝑏 ) ) |
38 |
37
|
rexbidv |
⊢ ( 𝑥 = 𝑎 → ( ∃ 𝑏 ∈ 𝐴 ∀ 𝑐 ∈ ( 𝐹 “ 𝑥 ) 𝑐 𝑅 𝑏 ↔ ∃ 𝑏 ∈ 𝐴 ∀ 𝑐 ∈ ( 𝐹 “ 𝑎 ) 𝑐 𝑅 𝑏 ) ) |
39 |
35 38
|
syl5bb |
⊢ ( 𝑥 = 𝑎 → ( ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 ↔ ∃ 𝑏 ∈ 𝐴 ∀ 𝑐 ∈ ( 𝐹 “ 𝑎 ) 𝑐 𝑅 𝑏 ) ) |
40 |
39
|
cbvrabv |
⊢ { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 } = { 𝑎 ∈ On ∣ ∃ 𝑏 ∈ 𝐴 ∀ 𝑐 ∈ ( 𝐹 “ 𝑎 ) 𝑐 𝑅 𝑏 } |
41 |
4 40
|
eqtri |
⊢ 𝑇 = { 𝑎 ∈ On ∣ ∃ 𝑏 ∈ 𝐴 ∀ 𝑐 ∈ ( 𝐹 “ 𝑎 ) 𝑐 𝑅 𝑏 } |
42 |
29 41
|
elrab2 |
⊢ ( 𝑇 ∈ 𝑇 ↔ ( 𝑇 ∈ On ∧ ∃ 𝑏 ∈ 𝐴 ∀ 𝑐 ∈ ( 𝐹 “ 𝑇 ) 𝑐 𝑅 𝑏 ) ) |
43 |
42
|
baib |
⊢ ( 𝑇 ∈ On → ( 𝑇 ∈ 𝑇 ↔ ∃ 𝑏 ∈ 𝐴 ∀ 𝑐 ∈ ( 𝐹 “ 𝑇 ) 𝑐 𝑅 𝑏 ) ) |
44 |
26 43
|
syl |
⊢ ( 𝜑 → ( 𝑇 ∈ 𝑇 ↔ ∃ 𝑏 ∈ 𝐴 ∀ 𝑐 ∈ ( 𝐹 “ 𝑇 ) 𝑐 𝑅 𝑏 ) ) |
45 |
14 44
|
mtbid |
⊢ ( 𝜑 → ¬ ∃ 𝑏 ∈ 𝐴 ∀ 𝑐 ∈ ( 𝐹 “ 𝑇 ) 𝑐 𝑅 𝑏 ) |
46 |
|
ralnex |
⊢ ( ∀ 𝑏 ∈ 𝐴 ¬ ∀ 𝑐 ∈ ( 𝐹 “ 𝑇 ) 𝑐 𝑅 𝑏 ↔ ¬ ∃ 𝑏 ∈ 𝐴 ∀ 𝑐 ∈ ( 𝐹 “ 𝑇 ) 𝑐 𝑅 𝑏 ) |
47 |
45 46
|
sylibr |
⊢ ( 𝜑 → ∀ 𝑏 ∈ 𝐴 ¬ ∀ 𝑐 ∈ ( 𝐹 “ 𝑇 ) 𝑐 𝑅 𝑏 ) |
48 |
47
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐴 ) → ¬ ∀ 𝑐 ∈ ( 𝐹 “ 𝑇 ) 𝑐 𝑅 𝑏 ) |
49 |
19
|
rneqd |
⊢ ( 𝜑 → ran 𝑂 = ran ( 𝐹 ↾ 𝑇 ) ) |
50 |
|
df-ima |
⊢ ( 𝐹 “ 𝑇 ) = ran ( 𝐹 ↾ 𝑇 ) |
51 |
49 50
|
eqtr4di |
⊢ ( 𝜑 → ran 𝑂 = ( 𝐹 “ 𝑇 ) ) |
52 |
51
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐴 ) → ran 𝑂 = ( 𝐹 “ 𝑇 ) ) |
53 |
52
|
raleqdv |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐴 ) → ( ∀ 𝑐 ∈ ran 𝑂 𝑐 𝑅 𝑏 ↔ ∀ 𝑐 ∈ ( 𝐹 “ 𝑇 ) 𝑐 𝑅 𝑏 ) ) |
54 |
10
|
ffund |
⊢ ( 𝜑 → Fun 𝑂 ) |
55 |
54
|
funfnd |
⊢ ( 𝜑 → 𝑂 Fn dom 𝑂 ) |
56 |
55
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐴 ) → 𝑂 Fn dom 𝑂 ) |
57 |
|
breq1 |
⊢ ( 𝑐 = ( 𝑂 ‘ 𝑚 ) → ( 𝑐 𝑅 𝑏 ↔ ( 𝑂 ‘ 𝑚 ) 𝑅 𝑏 ) ) |
58 |
57
|
ralrn |
⊢ ( 𝑂 Fn dom 𝑂 → ( ∀ 𝑐 ∈ ran 𝑂 𝑐 𝑅 𝑏 ↔ ∀ 𝑚 ∈ dom 𝑂 ( 𝑂 ‘ 𝑚 ) 𝑅 𝑏 ) ) |
59 |
56 58
|
syl |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐴 ) → ( ∀ 𝑐 ∈ ran 𝑂 𝑐 𝑅 𝑏 ↔ ∀ 𝑚 ∈ dom 𝑂 ( 𝑂 ‘ 𝑚 ) 𝑅 𝑏 ) ) |
60 |
53 59
|
bitr3d |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐴 ) → ( ∀ 𝑐 ∈ ( 𝐹 “ 𝑇 ) 𝑐 𝑅 𝑏 ↔ ∀ 𝑚 ∈ dom 𝑂 ( 𝑂 ‘ 𝑚 ) 𝑅 𝑏 ) ) |
61 |
48 60
|
mtbid |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐴 ) → ¬ ∀ 𝑚 ∈ dom 𝑂 ( 𝑂 ‘ 𝑚 ) 𝑅 𝑏 ) |
62 |
|
rexnal |
⊢ ( ∃ 𝑚 ∈ dom 𝑂 ¬ ( 𝑂 ‘ 𝑚 ) 𝑅 𝑏 ↔ ¬ ∀ 𝑚 ∈ dom 𝑂 ( 𝑂 ‘ 𝑚 ) 𝑅 𝑏 ) |
63 |
61 62
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐴 ) → ∃ 𝑚 ∈ dom 𝑂 ¬ ( 𝑂 ‘ 𝑚 ) 𝑅 𝑏 ) |
64 |
1 2 3 4 5 6 7
|
ordtypelem7 |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐴 ) ∧ 𝑚 ∈ dom 𝑂 ) → ( ( 𝑂 ‘ 𝑚 ) 𝑅 𝑏 ∨ 𝑏 ∈ ran 𝑂 ) ) |
65 |
64
|
ord |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ 𝐴 ) ∧ 𝑚 ∈ dom 𝑂 ) → ( ¬ ( 𝑂 ‘ 𝑚 ) 𝑅 𝑏 → 𝑏 ∈ ran 𝑂 ) ) |
66 |
65
|
rexlimdva |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐴 ) → ( ∃ 𝑚 ∈ dom 𝑂 ¬ ( 𝑂 ‘ 𝑚 ) 𝑅 𝑏 → 𝑏 ∈ ran 𝑂 ) ) |
67 |
63 66
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝐴 ) → 𝑏 ∈ ran 𝑂 ) |
68 |
11 67
|
eqelssd |
⊢ ( 𝜑 → ran 𝑂 = 𝐴 ) |
69 |
|
isoeq5 |
⊢ ( ran 𝑂 = 𝐴 → ( 𝑂 Isom E , 𝑅 ( dom 𝑂 , ran 𝑂 ) ↔ 𝑂 Isom E , 𝑅 ( dom 𝑂 , 𝐴 ) ) ) |
70 |
68 69
|
syl |
⊢ ( 𝜑 → ( 𝑂 Isom E , 𝑅 ( dom 𝑂 , ran 𝑂 ) ↔ 𝑂 Isom E , 𝑅 ( dom 𝑂 , 𝐴 ) ) ) |
71 |
9 70
|
mpbid |
⊢ ( 𝜑 → 𝑂 Isom E , 𝑅 ( dom 𝑂 , 𝐴 ) ) |