Step |
Hyp |
Ref |
Expression |
1 |
|
noinfdm.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 |
|
iffalse |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → 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 ) ) ) ) = ( 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 ) ) ) ) |
3 |
1 2
|
syl5eq |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → 𝑇 = ( 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 ) ) ) ) |
4 |
3
|
dmeqd |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = dom ( 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 ) ) ) ) |
5 |
|
iotaex |
⊢ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ∈ V |
6 |
|
eqid |
⊢ ( 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 ) ) ) = ( 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 ) ) ) |
7 |
5 6
|
dmmpti |
⊢ dom ( 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 ) ) ) = { 𝑦 ∣ ∃ 𝑢 ∈ 𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } |
8 |
|
eleq1w |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ dom 𝑢 ↔ 𝑧 ∈ dom 𝑢 ) ) |
9 |
|
suceq |
⊢ ( 𝑦 = 𝑧 → suc 𝑦 = suc 𝑧 ) |
10 |
9
|
reseq2d |
⊢ ( 𝑦 = 𝑧 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑢 ↾ suc 𝑧 ) ) |
11 |
9
|
reseq2d |
⊢ ( 𝑦 = 𝑧 → ( 𝑣 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑧 ) ) |
12 |
10 11
|
eqeq12d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ↔ ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) |
13 |
12
|
imbi2d |
⊢ ( 𝑦 = 𝑧 → ( ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ↔ ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) ) |
14 |
13
|
ralbidv |
⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ↔ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) ) |
15 |
8 14
|
anbi12d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) ↔ ( 𝑧 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) ) ) |
16 |
15
|
rexbidv |
⊢ ( 𝑦 = 𝑧 → ( ∃ 𝑢 ∈ 𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) ↔ ∃ 𝑢 ∈ 𝐵 ( 𝑧 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) ) ) |
17 |
|
dmeq |
⊢ ( 𝑢 = 𝑝 → dom 𝑢 = dom 𝑝 ) |
18 |
17
|
eleq2d |
⊢ ( 𝑢 = 𝑝 → ( 𝑧 ∈ dom 𝑢 ↔ 𝑧 ∈ dom 𝑝 ) ) |
19 |
|
breq1 |
⊢ ( 𝑢 = 𝑝 → ( 𝑢 <s 𝑣 ↔ 𝑝 <s 𝑣 ) ) |
20 |
19
|
notbid |
⊢ ( 𝑢 = 𝑝 → ( ¬ 𝑢 <s 𝑣 ↔ ¬ 𝑝 <s 𝑣 ) ) |
21 |
|
reseq1 |
⊢ ( 𝑢 = 𝑝 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑝 ↾ suc 𝑧 ) ) |
22 |
21
|
eqeq1d |
⊢ ( 𝑢 = 𝑝 → ( ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ↔ ( 𝑝 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) |
23 |
20 22
|
imbi12d |
⊢ ( 𝑢 = 𝑝 → ( ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ↔ ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) ) |
24 |
23
|
ralbidv |
⊢ ( 𝑢 = 𝑝 → ( ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ↔ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) ) |
25 |
|
breq2 |
⊢ ( 𝑣 = 𝑞 → ( 𝑝 <s 𝑣 ↔ 𝑝 <s 𝑞 ) ) |
26 |
25
|
notbid |
⊢ ( 𝑣 = 𝑞 → ( ¬ 𝑝 <s 𝑣 ↔ ¬ 𝑝 <s 𝑞 ) ) |
27 |
|
reseq1 |
⊢ ( 𝑣 = 𝑞 → ( 𝑣 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) |
28 |
27
|
eqeq2d |
⊢ ( 𝑣 = 𝑞 → ( ( 𝑝 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ↔ ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) |
29 |
26 28
|
imbi12d |
⊢ ( 𝑣 = 𝑞 → ( ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ↔ ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) ) |
30 |
29
|
cbvralvw |
⊢ ( ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ↔ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) |
31 |
24 30
|
bitrdi |
⊢ ( 𝑢 = 𝑝 → ( ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ↔ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) ) |
32 |
18 31
|
anbi12d |
⊢ ( 𝑢 = 𝑝 → ( ( 𝑧 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) ↔ ( 𝑧 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) ) ) |
33 |
32
|
cbvrexvw |
⊢ ( ∃ 𝑢 ∈ 𝐵 ( 𝑧 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑧 ) = ( 𝑣 ↾ suc 𝑧 ) ) ) ↔ ∃ 𝑝 ∈ 𝐵 ( 𝑧 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) ) |
34 |
16 33
|
bitrdi |
⊢ ( 𝑦 = 𝑧 → ( ∃ 𝑢 ∈ 𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) ↔ ∃ 𝑝 ∈ 𝐵 ( 𝑧 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) ) ) |
35 |
34
|
cbvabv |
⊢ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } = { 𝑧 ∣ ∃ 𝑝 ∈ 𝐵 ( 𝑧 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) } |
36 |
7 35
|
eqtri |
⊢ dom ( 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 ) ) ) = { 𝑧 ∣ ∃ 𝑝 ∈ 𝐵 ( 𝑧 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) } |
37 |
4 36
|
eqtrdi |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = { 𝑧 ∣ ∃ 𝑝 ∈ 𝐵 ( 𝑧 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc 𝑧 ) = ( 𝑞 ↾ suc 𝑧 ) ) ) } ) |