Step |
Hyp |
Ref |
Expression |
1 |
|
nosupres.1 |
⊢ 𝑆 = if ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 , ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { 〈 dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o 〉 } ) , ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) ) |
2 |
|
dmres |
⊢ dom ( 𝑆 ↾ suc 𝐺 ) = ( suc 𝐺 ∩ dom 𝑆 ) |
3 |
1
|
nosupno |
⊢ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) → 𝑆 ∈ No ) |
4 |
3
|
3ad2ant2 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝑆 ∈ No ) |
5 |
|
nodmord |
⊢ ( 𝑆 ∈ No → Ord dom 𝑆 ) |
6 |
4 5
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → Ord dom 𝑆 ) |
7 |
|
dmeq |
⊢ ( 𝑝 = 𝑈 → dom 𝑝 = dom 𝑈 ) |
8 |
7
|
eleq2d |
⊢ ( 𝑝 = 𝑈 → ( 𝐺 ∈ dom 𝑝 ↔ 𝐺 ∈ dom 𝑈 ) ) |
9 |
|
breq2 |
⊢ ( 𝑝 = 𝑈 → ( 𝑣 <s 𝑝 ↔ 𝑣 <s 𝑈 ) ) |
10 |
9
|
notbid |
⊢ ( 𝑝 = 𝑈 → ( ¬ 𝑣 <s 𝑝 ↔ ¬ 𝑣 <s 𝑈 ) ) |
11 |
|
reseq1 |
⊢ ( 𝑝 = 𝑈 → ( 𝑝 ↾ suc 𝐺 ) = ( 𝑈 ↾ suc 𝐺 ) ) |
12 |
11
|
eqeq1d |
⊢ ( 𝑝 = 𝑈 → ( ( 𝑝 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ↔ ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) |
13 |
10 12
|
imbi12d |
⊢ ( 𝑝 = 𝑈 → ( ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ↔ ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) |
14 |
13
|
ralbidv |
⊢ ( 𝑝 = 𝑈 → ( ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ↔ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) |
15 |
8 14
|
anbi12d |
⊢ ( 𝑝 = 𝑈 → ( ( 𝐺 ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ↔ ( 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ) |
16 |
15
|
rspcev |
⊢ ( ( 𝑈 ∈ 𝐴 ∧ ( 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ∃ 𝑝 ∈ 𝐴 ( 𝐺 ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) |
17 |
16
|
3impb |
⊢ ( ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) → ∃ 𝑝 ∈ 𝐴 ( 𝐺 ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) |
18 |
|
dmeq |
⊢ ( 𝑢 = 𝑝 → dom 𝑢 = dom 𝑝 ) |
19 |
18
|
eleq2d |
⊢ ( 𝑢 = 𝑝 → ( 𝐺 ∈ dom 𝑢 ↔ 𝐺 ∈ dom 𝑝 ) ) |
20 |
|
breq2 |
⊢ ( 𝑢 = 𝑝 → ( 𝑣 <s 𝑢 ↔ 𝑣 <s 𝑝 ) ) |
21 |
20
|
notbid |
⊢ ( 𝑢 = 𝑝 → ( ¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑝 ) ) |
22 |
|
reseq1 |
⊢ ( 𝑢 = 𝑝 → ( 𝑢 ↾ suc 𝐺 ) = ( 𝑝 ↾ suc 𝐺 ) ) |
23 |
22
|
eqeq1d |
⊢ ( 𝑢 = 𝑝 → ( ( 𝑢 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ↔ ( 𝑝 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) |
24 |
21 23
|
imbi12d |
⊢ ( 𝑢 = 𝑝 → ( ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ↔ ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) |
25 |
24
|
ralbidv |
⊢ ( 𝑢 = 𝑝 → ( ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ↔ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) |
26 |
19 25
|
anbi12d |
⊢ ( 𝑢 = 𝑝 → ( ( 𝐺 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ↔ ( 𝐺 ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ) |
27 |
26
|
cbvrexvw |
⊢ ( ∃ 𝑢 ∈ 𝐴 ( 𝐺 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ↔ ∃ 𝑝 ∈ 𝐴 ( 𝐺 ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) |
28 |
17 27
|
sylibr |
⊢ ( ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) → ∃ 𝑢 ∈ 𝐴 ( 𝐺 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) |
29 |
|
eleq1 |
⊢ ( 𝑦 = 𝐺 → ( 𝑦 ∈ dom 𝑢 ↔ 𝐺 ∈ dom 𝑢 ) ) |
30 |
|
suceq |
⊢ ( 𝑦 = 𝐺 → suc 𝑦 = suc 𝐺 ) |
31 |
30
|
reseq2d |
⊢ ( 𝑦 = 𝐺 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑢 ↾ suc 𝐺 ) ) |
32 |
30
|
reseq2d |
⊢ ( 𝑦 = 𝐺 → ( 𝑣 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝐺 ) ) |
33 |
31 32
|
eqeq12d |
⊢ ( 𝑦 = 𝐺 → ( ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ↔ ( 𝑢 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) |
34 |
33
|
imbi2d |
⊢ ( 𝑦 = 𝐺 → ( ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ↔ ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) |
35 |
34
|
ralbidv |
⊢ ( 𝑦 = 𝐺 → ( ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ↔ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) |
36 |
29 35
|
anbi12d |
⊢ ( 𝑦 = 𝐺 → ( ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) ↔ ( 𝐺 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ) |
37 |
36
|
rexbidv |
⊢ ( 𝑦 = 𝐺 → ( ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) ↔ ∃ 𝑢 ∈ 𝐴 ( 𝐺 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ) |
38 |
37
|
elabg |
⊢ ( 𝐺 ∈ dom 𝑈 → ( 𝐺 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↔ ∃ 𝑢 ∈ 𝐴 ( 𝐺 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ) |
39 |
38
|
3ad2ant2 |
⊢ ( ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) → ( 𝐺 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↔ ∃ 𝑢 ∈ 𝐴 ( 𝐺 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ) |
40 |
28 39
|
mpbird |
⊢ ( ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) → 𝐺 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ) |
41 |
40
|
3ad2ant3 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝐺 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ) |
42 |
|
iffalse |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → if ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 , ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { 〈 dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o 〉 } ) , ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) ) = ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) ) |
43 |
1 42
|
syl5eq |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → 𝑆 = ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) ) |
44 |
43
|
dmeqd |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) ) |
45 |
|
iotaex |
⊢ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ∈ V |
46 |
|
eqid |
⊢ ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) = ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) |
47 |
45 46
|
dmmpti |
⊢ dom ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) = { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } |
48 |
44 47
|
eqtrdi |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ) |
49 |
48
|
3ad2ant1 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → dom 𝑆 = { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ) |
50 |
41 49
|
eleqtrrd |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝐺 ∈ dom 𝑆 ) |
51 |
|
ordsucss |
⊢ ( Ord dom 𝑆 → ( 𝐺 ∈ dom 𝑆 → suc 𝐺 ⊆ dom 𝑆 ) ) |
52 |
6 50 51
|
sylc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → suc 𝐺 ⊆ dom 𝑆 ) |
53 |
|
df-ss |
⊢ ( suc 𝐺 ⊆ dom 𝑆 ↔ ( suc 𝐺 ∩ dom 𝑆 ) = suc 𝐺 ) |
54 |
52 53
|
sylib |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( suc 𝐺 ∩ dom 𝑆 ) = suc 𝐺 ) |
55 |
2 54
|
syl5eq |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → dom ( 𝑆 ↾ suc 𝐺 ) = suc 𝐺 ) |
56 |
|
dmres |
⊢ dom ( 𝑈 ↾ suc 𝐺 ) = ( suc 𝐺 ∩ dom 𝑈 ) |
57 |
|
simp2l |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝐴 ⊆ No ) |
58 |
|
simp31 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝑈 ∈ 𝐴 ) |
59 |
57 58
|
sseldd |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝑈 ∈ No ) |
60 |
|
nodmord |
⊢ ( 𝑈 ∈ No → Ord dom 𝑈 ) |
61 |
59 60
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → Ord dom 𝑈 ) |
62 |
|
simp32 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝐺 ∈ dom 𝑈 ) |
63 |
|
ordsucss |
⊢ ( Ord dom 𝑈 → ( 𝐺 ∈ dom 𝑈 → suc 𝐺 ⊆ dom 𝑈 ) ) |
64 |
61 62 63
|
sylc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → suc 𝐺 ⊆ dom 𝑈 ) |
65 |
|
df-ss |
⊢ ( suc 𝐺 ⊆ dom 𝑈 ↔ ( suc 𝐺 ∩ dom 𝑈 ) = suc 𝐺 ) |
66 |
64 65
|
sylib |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( suc 𝐺 ∩ dom 𝑈 ) = suc 𝐺 ) |
67 |
56 66
|
syl5eq |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → dom ( 𝑈 ↾ suc 𝐺 ) = suc 𝐺 ) |
68 |
55 67
|
eqtr4d |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → dom ( 𝑆 ↾ suc 𝐺 ) = dom ( 𝑈 ↾ suc 𝐺 ) ) |
69 |
55
|
eleq2d |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( 𝑎 ∈ dom ( 𝑆 ↾ suc 𝐺 ) ↔ 𝑎 ∈ suc 𝐺 ) ) |
70 |
|
simpl1 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) |
71 |
|
simpl2 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ) |
72 |
|
simpl31 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → 𝑈 ∈ 𝐴 ) |
73 |
64
|
sselda |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → 𝑎 ∈ dom 𝑈 ) |
74 |
|
nodmon |
⊢ ( 𝑈 ∈ No → dom 𝑈 ∈ On ) |
75 |
59 74
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → dom 𝑈 ∈ On ) |
76 |
|
onelon |
⊢ ( ( dom 𝑈 ∈ On ∧ 𝐺 ∈ dom 𝑈 ) → 𝐺 ∈ On ) |
77 |
75 62 76
|
syl2anc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝐺 ∈ On ) |
78 |
|
eloni |
⊢ ( 𝐺 ∈ On → Ord 𝐺 ) |
79 |
77 78
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → Ord 𝐺 ) |
80 |
|
ordsuc |
⊢ ( Ord 𝐺 ↔ Ord suc 𝐺 ) |
81 |
79 80
|
sylib |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → Ord suc 𝐺 ) |
82 |
|
ordsucss |
⊢ ( Ord suc 𝐺 → ( 𝑎 ∈ suc 𝐺 → suc 𝑎 ⊆ suc 𝐺 ) ) |
83 |
81 82
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( 𝑎 ∈ suc 𝐺 → suc 𝑎 ⊆ suc 𝐺 ) ) |
84 |
83
|
imp |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → suc 𝑎 ⊆ suc 𝐺 ) |
85 |
|
simpl33 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) |
86 |
|
reseq1 |
⊢ ( ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) → ( ( 𝑈 ↾ suc 𝐺 ) ↾ suc 𝑎 ) = ( ( 𝑣 ↾ suc 𝐺 ) ↾ suc 𝑎 ) ) |
87 |
|
resabs1 |
⊢ ( suc 𝑎 ⊆ suc 𝐺 → ( ( 𝑈 ↾ suc 𝐺 ) ↾ suc 𝑎 ) = ( 𝑈 ↾ suc 𝑎 ) ) |
88 |
|
resabs1 |
⊢ ( suc 𝑎 ⊆ suc 𝐺 → ( ( 𝑣 ↾ suc 𝐺 ) ↾ suc 𝑎 ) = ( 𝑣 ↾ suc 𝑎 ) ) |
89 |
87 88
|
eqeq12d |
⊢ ( suc 𝑎 ⊆ suc 𝐺 → ( ( ( 𝑈 ↾ suc 𝐺 ) ↾ suc 𝑎 ) = ( ( 𝑣 ↾ suc 𝐺 ) ↾ suc 𝑎 ) ↔ ( 𝑈 ↾ suc 𝑎 ) = ( 𝑣 ↾ suc 𝑎 ) ) ) |
90 |
86 89
|
syl5ib |
⊢ ( suc 𝑎 ⊆ suc 𝐺 → ( ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) → ( 𝑈 ↾ suc 𝑎 ) = ( 𝑣 ↾ suc 𝑎 ) ) ) |
91 |
90
|
imim2d |
⊢ ( suc 𝑎 ⊆ suc 𝐺 → ( ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) → ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝑎 ) = ( 𝑣 ↾ suc 𝑎 ) ) ) ) |
92 |
91
|
ralimdv |
⊢ ( suc 𝑎 ⊆ suc 𝐺 → ( ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) → ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝑎 ) = ( 𝑣 ↾ suc 𝑎 ) ) ) ) |
93 |
84 85 92
|
sylc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝑎 ) = ( 𝑣 ↾ suc 𝑎 ) ) ) |
94 |
1
|
nosupfv |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑎 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝑎 ) = ( 𝑣 ↾ suc 𝑎 ) ) ) ) → ( 𝑆 ‘ 𝑎 ) = ( 𝑈 ‘ 𝑎 ) ) |
95 |
70 71 72 73 93 94
|
syl113anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ( 𝑆 ‘ 𝑎 ) = ( 𝑈 ‘ 𝑎 ) ) |
96 |
|
simpr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → 𝑎 ∈ suc 𝐺 ) |
97 |
96
|
fvresd |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ( ( 𝑆 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( 𝑆 ‘ 𝑎 ) ) |
98 |
96
|
fvresd |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ( ( 𝑈 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( 𝑈 ‘ 𝑎 ) ) |
99 |
95 97 98
|
3eqtr4d |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ( ( 𝑆 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( ( 𝑈 ↾ suc 𝐺 ) ‘ 𝑎 ) ) |
100 |
99
|
ex |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( 𝑎 ∈ suc 𝐺 → ( ( 𝑆 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( ( 𝑈 ↾ suc 𝐺 ) ‘ 𝑎 ) ) ) |
101 |
69 100
|
sylbid |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( 𝑎 ∈ dom ( 𝑆 ↾ suc 𝐺 ) → ( ( 𝑆 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( ( 𝑈 ↾ suc 𝐺 ) ‘ 𝑎 ) ) ) |
102 |
101
|
ralrimiv |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ∀ 𝑎 ∈ dom ( 𝑆 ↾ suc 𝐺 ) ( ( 𝑆 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( ( 𝑈 ↾ suc 𝐺 ) ‘ 𝑎 ) ) |
103 |
|
nofun |
⊢ ( 𝑆 ∈ No → Fun 𝑆 ) |
104 |
|
funres |
⊢ ( Fun 𝑆 → Fun ( 𝑆 ↾ suc 𝐺 ) ) |
105 |
4 103 104
|
3syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → Fun ( 𝑆 ↾ suc 𝐺 ) ) |
106 |
|
nofun |
⊢ ( 𝑈 ∈ No → Fun 𝑈 ) |
107 |
|
funres |
⊢ ( Fun 𝑈 → Fun ( 𝑈 ↾ suc 𝐺 ) ) |
108 |
59 106 107
|
3syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → Fun ( 𝑈 ↾ suc 𝐺 ) ) |
109 |
|
eqfunfv |
⊢ ( ( Fun ( 𝑆 ↾ suc 𝐺 ) ∧ Fun ( 𝑈 ↾ suc 𝐺 ) ) → ( ( 𝑆 ↾ suc 𝐺 ) = ( 𝑈 ↾ suc 𝐺 ) ↔ ( dom ( 𝑆 ↾ suc 𝐺 ) = dom ( 𝑈 ↾ suc 𝐺 ) ∧ ∀ 𝑎 ∈ dom ( 𝑆 ↾ suc 𝐺 ) ( ( 𝑆 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( ( 𝑈 ↾ suc 𝐺 ) ‘ 𝑎 ) ) ) ) |
110 |
105 108 109
|
syl2anc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( ( 𝑆 ↾ suc 𝐺 ) = ( 𝑈 ↾ suc 𝐺 ) ↔ ( dom ( 𝑆 ↾ suc 𝐺 ) = dom ( 𝑈 ↾ suc 𝐺 ) ∧ ∀ 𝑎 ∈ dom ( 𝑆 ↾ suc 𝐺 ) ( ( 𝑆 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( ( 𝑈 ↾ suc 𝐺 ) ‘ 𝑎 ) ) ) ) |
111 |
68 102 110
|
mpbir2and |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑈 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( 𝑆 ↾ suc 𝐺 ) = ( 𝑈 ↾ suc 𝐺 ) ) |