Step |
Hyp |
Ref |
Expression |
1 |
|
noinfres.1 |
⊢ 𝑇 = if ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 , ( ( ℩ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) ∪ { 〈 dom ( ℩ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) , 1o 〉 } ) , ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) ) |
2 |
|
dmres |
⊢ dom ( 𝑇 ↾ suc 𝐺 ) = ( suc 𝐺 ∩ dom 𝑇 ) |
3 |
1
|
noinfno |
⊢ ( ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) → 𝑇 ∈ No ) |
4 |
3
|
3ad2ant2 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝑇 ∈ No ) |
5 |
|
nodmord |
⊢ ( 𝑇 ∈ No → Ord dom 𝑇 ) |
6 |
4 5
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → Ord dom 𝑇 ) |
7 |
|
simp31 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝑈 ∈ 𝐵 ) |
8 |
|
simp32 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝐺 ∈ dom 𝑈 ) |
9 |
|
simp33 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) |
10 |
|
dmeq |
⊢ ( 𝑏 = 𝑈 → dom 𝑏 = dom 𝑈 ) |
11 |
10
|
eleq2d |
⊢ ( 𝑏 = 𝑈 → ( 𝐺 ∈ dom 𝑏 ↔ 𝐺 ∈ dom 𝑈 ) ) |
12 |
|
breq1 |
⊢ ( 𝑏 = 𝑈 → ( 𝑏 <s 𝑐 ↔ 𝑈 <s 𝑐 ) ) |
13 |
12
|
notbid |
⊢ ( 𝑏 = 𝑈 → ( ¬ 𝑏 <s 𝑐 ↔ ¬ 𝑈 <s 𝑐 ) ) |
14 |
|
reseq1 |
⊢ ( 𝑏 = 𝑈 → ( 𝑏 ↾ suc 𝐺 ) = ( 𝑈 ↾ suc 𝐺 ) ) |
15 |
14
|
eqeq1d |
⊢ ( 𝑏 = 𝑈 → ( ( 𝑏 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ↔ ( 𝑈 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ) |
16 |
13 15
|
imbi12d |
⊢ ( 𝑏 = 𝑈 → ( ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ↔ ( ¬ 𝑈 <s 𝑐 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ) ) |
17 |
16
|
ralbidv |
⊢ ( 𝑏 = 𝑈 → ( ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ↔ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑈 <s 𝑐 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ) ) |
18 |
|
breq2 |
⊢ ( 𝑐 = 𝑣 → ( 𝑈 <s 𝑐 ↔ 𝑈 <s 𝑣 ) ) |
19 |
18
|
notbid |
⊢ ( 𝑐 = 𝑣 → ( ¬ 𝑈 <s 𝑐 ↔ ¬ 𝑈 <s 𝑣 ) ) |
20 |
|
reseq1 |
⊢ ( 𝑐 = 𝑣 → ( 𝑐 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) |
21 |
20
|
eqeq2d |
⊢ ( 𝑐 = 𝑣 → ( ( 𝑈 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ↔ ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) |
22 |
19 21
|
imbi12d |
⊢ ( 𝑐 = 𝑣 → ( ( ¬ 𝑈 <s 𝑐 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ↔ ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) |
23 |
22
|
cbvralvw |
⊢ ( ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑈 <s 𝑐 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ↔ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) |
24 |
17 23
|
bitrdi |
⊢ ( 𝑏 = 𝑈 → ( ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ↔ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) |
25 |
11 24
|
anbi12d |
⊢ ( 𝑏 = 𝑈 → ( ( 𝐺 ∈ dom 𝑏 ∧ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ) ↔ ( 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ) |
26 |
25
|
rspcev |
⊢ ( ( 𝑈 ∈ 𝐵 ∧ ( 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ∃ 𝑏 ∈ 𝐵 ( 𝐺 ∈ dom 𝑏 ∧ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ) ) |
27 |
7 8 9 26
|
syl12anc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ∃ 𝑏 ∈ 𝐵 ( 𝐺 ∈ dom 𝑏 ∧ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ) ) |
28 |
|
eleq1 |
⊢ ( 𝑎 = 𝐺 → ( 𝑎 ∈ dom 𝑏 ↔ 𝐺 ∈ dom 𝑏 ) ) |
29 |
|
suceq |
⊢ ( 𝑎 = 𝐺 → suc 𝑎 = suc 𝐺 ) |
30 |
29
|
reseq2d |
⊢ ( 𝑎 = 𝐺 → ( 𝑏 ↾ suc 𝑎 ) = ( 𝑏 ↾ suc 𝐺 ) ) |
31 |
29
|
reseq2d |
⊢ ( 𝑎 = 𝐺 → ( 𝑐 ↾ suc 𝑎 ) = ( 𝑐 ↾ suc 𝐺 ) ) |
32 |
30 31
|
eqeq12d |
⊢ ( 𝑎 = 𝐺 → ( ( 𝑏 ↾ suc 𝑎 ) = ( 𝑐 ↾ suc 𝑎 ) ↔ ( 𝑏 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ) |
33 |
32
|
imbi2d |
⊢ ( 𝑎 = 𝐺 → ( ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝑎 ) = ( 𝑐 ↾ suc 𝑎 ) ) ↔ ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ) ) |
34 |
33
|
ralbidv |
⊢ ( 𝑎 = 𝐺 → ( ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝑎 ) = ( 𝑐 ↾ suc 𝑎 ) ) ↔ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ) ) |
35 |
28 34
|
anbi12d |
⊢ ( 𝑎 = 𝐺 → ( ( 𝑎 ∈ dom 𝑏 ∧ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝑎 ) = ( 𝑐 ↾ suc 𝑎 ) ) ) ↔ ( 𝐺 ∈ dom 𝑏 ∧ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ) ) ) |
36 |
35
|
rexbidv |
⊢ ( 𝑎 = 𝐺 → ( ∃ 𝑏 ∈ 𝐵 ( 𝑎 ∈ dom 𝑏 ∧ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝑎 ) = ( 𝑐 ↾ suc 𝑎 ) ) ) ↔ ∃ 𝑏 ∈ 𝐵 ( 𝐺 ∈ dom 𝑏 ∧ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ) ) ) |
37 |
36
|
elabg |
⊢ ( 𝐺 ∈ dom 𝑈 → ( 𝐺 ∈ { 𝑎 ∣ ∃ 𝑏 ∈ 𝐵 ( 𝑎 ∈ dom 𝑏 ∧ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝑎 ) = ( 𝑐 ↾ suc 𝑎 ) ) ) } ↔ ∃ 𝑏 ∈ 𝐵 ( 𝐺 ∈ dom 𝑏 ∧ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ) ) ) |
38 |
8 37
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( 𝐺 ∈ { 𝑎 ∣ ∃ 𝑏 ∈ 𝐵 ( 𝑎 ∈ dom 𝑏 ∧ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝑎 ) = ( 𝑐 ↾ suc 𝑎 ) ) ) } ↔ ∃ 𝑏 ∈ 𝐵 ( 𝐺 ∈ dom 𝑏 ∧ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝐺 ) = ( 𝑐 ↾ suc 𝐺 ) ) ) ) ) |
39 |
27 38
|
mpbird |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝐺 ∈ { 𝑎 ∣ ∃ 𝑏 ∈ 𝐵 ( 𝑎 ∈ dom 𝑏 ∧ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝑎 ) = ( 𝑐 ↾ suc 𝑎 ) ) ) } ) |
40 |
1
|
noinfdm |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = { 𝑎 ∣ ∃ 𝑏 ∈ 𝐵 ( 𝑎 ∈ dom 𝑏 ∧ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝑎 ) = ( 𝑐 ↾ suc 𝑎 ) ) ) } ) |
41 |
40
|
3ad2ant1 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → dom 𝑇 = { 𝑎 ∣ ∃ 𝑏 ∈ 𝐵 ( 𝑎 ∈ dom 𝑏 ∧ ∀ 𝑐 ∈ 𝐵 ( ¬ 𝑏 <s 𝑐 → ( 𝑏 ↾ suc 𝑎 ) = ( 𝑐 ↾ suc 𝑎 ) ) ) } ) |
42 |
39 41
|
eleqtrrd |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝐺 ∈ dom 𝑇 ) |
43 |
|
ordsucss |
⊢ ( Ord dom 𝑇 → ( 𝐺 ∈ dom 𝑇 → suc 𝐺 ⊆ dom 𝑇 ) ) |
44 |
6 42 43
|
sylc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → suc 𝐺 ⊆ dom 𝑇 ) |
45 |
|
df-ss |
⊢ ( suc 𝐺 ⊆ dom 𝑇 ↔ ( suc 𝐺 ∩ dom 𝑇 ) = suc 𝐺 ) |
46 |
44 45
|
sylib |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( suc 𝐺 ∩ dom 𝑇 ) = suc 𝐺 ) |
47 |
2 46
|
syl5eq |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → dom ( 𝑇 ↾ suc 𝐺 ) = suc 𝐺 ) |
48 |
|
dmres |
⊢ dom ( 𝑈 ↾ suc 𝐺 ) = ( suc 𝐺 ∩ dom 𝑈 ) |
49 |
|
simp2l |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝐵 ⊆ No ) |
50 |
49 7
|
sseldd |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → 𝑈 ∈ No ) |
51 |
|
nodmon |
⊢ ( 𝑈 ∈ No → dom 𝑈 ∈ On ) |
52 |
50 51
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → dom 𝑈 ∈ On ) |
53 |
|
eloni |
⊢ ( dom 𝑈 ∈ On → Ord dom 𝑈 ) |
54 |
52 53
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → Ord dom 𝑈 ) |
55 |
|
ordsucss |
⊢ ( Ord dom 𝑈 → ( 𝐺 ∈ dom 𝑈 → suc 𝐺 ⊆ dom 𝑈 ) ) |
56 |
54 8 55
|
sylc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → suc 𝐺 ⊆ dom 𝑈 ) |
57 |
|
df-ss |
⊢ ( suc 𝐺 ⊆ dom 𝑈 ↔ ( suc 𝐺 ∩ dom 𝑈 ) = suc 𝐺 ) |
58 |
56 57
|
sylib |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( suc 𝐺 ∩ dom 𝑈 ) = suc 𝐺 ) |
59 |
48 58
|
syl5eq |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → dom ( 𝑈 ↾ suc 𝐺 ) = suc 𝐺 ) |
60 |
47 59
|
eqtr4d |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → dom ( 𝑇 ↾ suc 𝐺 ) = dom ( 𝑈 ↾ suc 𝐺 ) ) |
61 |
47
|
eleq2d |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( 𝑎 ∈ dom ( 𝑇 ↾ suc 𝐺 ) ↔ 𝑎 ∈ suc 𝐺 ) ) |
62 |
|
simpl1 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) |
63 |
|
simpl2 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ) |
64 |
|
simpl31 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → 𝑈 ∈ 𝐵 ) |
65 |
56
|
sselda |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → 𝑎 ∈ dom 𝑈 ) |
66 |
50
|
adantr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → 𝑈 ∈ No ) |
67 |
66 51
|
syl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → dom 𝑈 ∈ On ) |
68 |
|
simpl32 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → 𝐺 ∈ dom 𝑈 ) |
69 |
|
onelon |
⊢ ( ( dom 𝑈 ∈ On ∧ 𝐺 ∈ dom 𝑈 ) → 𝐺 ∈ On ) |
70 |
67 68 69
|
syl2anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → 𝐺 ∈ On ) |
71 |
|
sucelon |
⊢ ( 𝐺 ∈ On ↔ suc 𝐺 ∈ On ) |
72 |
70 71
|
sylib |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → suc 𝐺 ∈ On ) |
73 |
|
eloni |
⊢ ( suc 𝐺 ∈ On → Ord suc 𝐺 ) |
74 |
72 73
|
syl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → Ord suc 𝐺 ) |
75 |
|
simpr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → 𝑎 ∈ suc 𝐺 ) |
76 |
|
ordsucss |
⊢ ( Ord suc 𝐺 → ( 𝑎 ∈ suc 𝐺 → suc 𝑎 ⊆ suc 𝐺 ) ) |
77 |
74 75 76
|
sylc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → suc 𝑎 ⊆ suc 𝐺 ) |
78 |
|
simpl33 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) |
79 |
|
reseq1 |
⊢ ( ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) → ( ( 𝑈 ↾ suc 𝐺 ) ↾ suc 𝑎 ) = ( ( 𝑣 ↾ suc 𝐺 ) ↾ suc 𝑎 ) ) |
80 |
|
resabs1 |
⊢ ( suc 𝑎 ⊆ suc 𝐺 → ( ( 𝑈 ↾ suc 𝐺 ) ↾ suc 𝑎 ) = ( 𝑈 ↾ suc 𝑎 ) ) |
81 |
|
resabs1 |
⊢ ( suc 𝑎 ⊆ suc 𝐺 → ( ( 𝑣 ↾ suc 𝐺 ) ↾ suc 𝑎 ) = ( 𝑣 ↾ suc 𝑎 ) ) |
82 |
80 81
|
eqeq12d |
⊢ ( suc 𝑎 ⊆ suc 𝐺 → ( ( ( 𝑈 ↾ suc 𝐺 ) ↾ suc 𝑎 ) = ( ( 𝑣 ↾ suc 𝐺 ) ↾ suc 𝑎 ) ↔ ( 𝑈 ↾ suc 𝑎 ) = ( 𝑣 ↾ suc 𝑎 ) ) ) |
83 |
79 82
|
syl5ib |
⊢ ( suc 𝑎 ⊆ suc 𝐺 → ( ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) → ( 𝑈 ↾ suc 𝑎 ) = ( 𝑣 ↾ suc 𝑎 ) ) ) |
84 |
83
|
imim2d |
⊢ ( suc 𝑎 ⊆ suc 𝐺 → ( ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) → ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝑎 ) = ( 𝑣 ↾ suc 𝑎 ) ) ) ) |
85 |
84
|
ralimdv |
⊢ ( suc 𝑎 ⊆ suc 𝐺 → ( ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) → ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝑎 ) = ( 𝑣 ↾ suc 𝑎 ) ) ) ) |
86 |
77 78 85
|
sylc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝑎 ) = ( 𝑣 ↾ suc 𝑎 ) ) ) |
87 |
1
|
noinffv |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝑎 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝑎 ) = ( 𝑣 ↾ suc 𝑎 ) ) ) ) → ( 𝑇 ‘ 𝑎 ) = ( 𝑈 ‘ 𝑎 ) ) |
88 |
62 63 64 65 86 87
|
syl113anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ( 𝑇 ‘ 𝑎 ) = ( 𝑈 ‘ 𝑎 ) ) |
89 |
75
|
fvresd |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ( ( 𝑇 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( 𝑇 ‘ 𝑎 ) ) |
90 |
75
|
fvresd |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ( ( 𝑈 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( 𝑈 ‘ 𝑎 ) ) |
91 |
88 89 90
|
3eqtr4d |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) ∧ 𝑎 ∈ suc 𝐺 ) → ( ( 𝑇 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( ( 𝑈 ↾ suc 𝐺 ) ‘ 𝑎 ) ) |
92 |
91
|
ex |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( 𝑎 ∈ suc 𝐺 → ( ( 𝑇 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( ( 𝑈 ↾ suc 𝐺 ) ‘ 𝑎 ) ) ) |
93 |
61 92
|
sylbid |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( 𝑎 ∈ dom ( 𝑇 ↾ suc 𝐺 ) → ( ( 𝑇 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( ( 𝑈 ↾ suc 𝐺 ) ‘ 𝑎 ) ) ) |
94 |
93
|
ralrimiv |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ∀ 𝑎 ∈ dom ( 𝑇 ↾ suc 𝐺 ) ( ( 𝑇 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( ( 𝑈 ↾ suc 𝐺 ) ‘ 𝑎 ) ) |
95 |
|
nofun |
⊢ ( 𝑇 ∈ No → Fun 𝑇 ) |
96 |
95
|
funresd |
⊢ ( 𝑇 ∈ No → Fun ( 𝑇 ↾ suc 𝐺 ) ) |
97 |
4 96
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → Fun ( 𝑇 ↾ suc 𝐺 ) ) |
98 |
|
nofun |
⊢ ( 𝑈 ∈ No → Fun 𝑈 ) |
99 |
98
|
funresd |
⊢ ( 𝑈 ∈ No → Fun ( 𝑈 ↾ suc 𝐺 ) ) |
100 |
50 99
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → Fun ( 𝑈 ↾ suc 𝐺 ) ) |
101 |
|
eqfunfv |
⊢ ( ( Fun ( 𝑇 ↾ suc 𝐺 ) ∧ Fun ( 𝑈 ↾ suc 𝐺 ) ) → ( ( 𝑇 ↾ suc 𝐺 ) = ( 𝑈 ↾ suc 𝐺 ) ↔ ( dom ( 𝑇 ↾ suc 𝐺 ) = dom ( 𝑈 ↾ suc 𝐺 ) ∧ ∀ 𝑎 ∈ dom ( 𝑇 ↾ suc 𝐺 ) ( ( 𝑇 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( ( 𝑈 ↾ suc 𝐺 ) ‘ 𝑎 ) ) ) ) |
102 |
97 100 101
|
syl2anc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( ( 𝑇 ↾ suc 𝐺 ) = ( 𝑈 ↾ suc 𝐺 ) ↔ ( dom ( 𝑇 ↾ suc 𝐺 ) = dom ( 𝑈 ↾ suc 𝐺 ) ∧ ∀ 𝑎 ∈ dom ( 𝑇 ↾ suc 𝐺 ) ( ( 𝑇 ↾ suc 𝐺 ) ‘ 𝑎 ) = ( ( 𝑈 ↾ suc 𝐺 ) ‘ 𝑎 ) ) ) ) |
103 |
60 94 102
|
mpbir2and |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑈 <s 𝑣 → ( 𝑈 ↾ suc 𝐺 ) = ( 𝑣 ↾ suc 𝐺 ) ) ) ) → ( 𝑇 ↾ suc 𝐺 ) = ( 𝑈 ↾ suc 𝐺 ) ) |