Step |
Hyp |
Ref |
Expression |
1 |
|
madeval |
⊢ ( 𝐴 ∈ On → ( M ‘ 𝐴 ) = ( |s “ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ) ) |
2 |
|
scutcl |
⊢ ( 𝑎 <<s 𝑏 → ( 𝑎 |s 𝑏 ) ∈ No ) |
3 |
|
eleq1 |
⊢ ( ( 𝑎 |s 𝑏 ) = 𝑥 → ( ( 𝑎 |s 𝑏 ) ∈ No ↔ 𝑥 ∈ No ) ) |
4 |
3
|
biimpd |
⊢ ( ( 𝑎 |s 𝑏 ) = 𝑥 → ( ( 𝑎 |s 𝑏 ) ∈ No → 𝑥 ∈ No ) ) |
5 |
2 4
|
mpan9 |
⊢ ( ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) → 𝑥 ∈ No ) |
6 |
5
|
rexlimivw |
⊢ ( ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) → 𝑥 ∈ No ) |
7 |
6
|
rexlimivw |
⊢ ( ∃ 𝑎 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) → 𝑥 ∈ No ) |
8 |
7
|
pm4.71ri |
⊢ ( ∃ 𝑎 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ↔ ( 𝑥 ∈ No ∧ ∃ 𝑎 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ) ) |
9 |
8
|
abbii |
⊢ { 𝑥 ∣ ∃ 𝑎 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) } = { 𝑥 ∣ ( 𝑥 ∈ No ∧ ∃ 𝑎 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ) } |
10 |
|
eleq1 |
⊢ ( 𝑦 = 〈 𝑎 , 𝑏 〉 → ( 𝑦 ∈ <<s ↔ 〈 𝑎 , 𝑏 〉 ∈ <<s ) ) |
11 |
|
breq1 |
⊢ ( 𝑦 = 〈 𝑎 , 𝑏 〉 → ( 𝑦 |s 𝑥 ↔ 〈 𝑎 , 𝑏 〉 |s 𝑥 ) ) |
12 |
10 11
|
anbi12d |
⊢ ( 𝑦 = 〈 𝑎 , 𝑏 〉 → ( ( 𝑦 ∈ <<s ∧ 𝑦 |s 𝑥 ) ↔ ( 〈 𝑎 , 𝑏 〉 ∈ <<s ∧ 〈 𝑎 , 𝑏 〉 |s 𝑥 ) ) ) |
13 |
12
|
rexxp |
⊢ ( ∃ 𝑦 ∈ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ( 𝑦 ∈ <<s ∧ 𝑦 |s 𝑥 ) ↔ ∃ 𝑎 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 〈 𝑎 , 𝑏 〉 ∈ <<s ∧ 〈 𝑎 , 𝑏 〉 |s 𝑥 ) ) |
14 |
|
imaindm |
⊢ ( |s “ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ) = ( |s “ ( ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∩ dom |s ) ) |
15 |
|
dmscut |
⊢ dom |s = <<s |
16 |
15
|
ineq2i |
⊢ ( ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∩ dom |s ) = ( ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∩ <<s ) |
17 |
16
|
imaeq2i |
⊢ ( |s “ ( ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∩ dom |s ) ) = ( |s “ ( ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∩ <<s ) ) |
18 |
14 17
|
eqtri |
⊢ ( |s “ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ) = ( |s “ ( ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∩ <<s ) ) |
19 |
18
|
eleq2i |
⊢ ( 𝑥 ∈ ( |s “ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ) ↔ 𝑥 ∈ ( |s “ ( ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∩ <<s ) ) ) |
20 |
|
vex |
⊢ 𝑥 ∈ V |
21 |
20
|
elima |
⊢ ( 𝑥 ∈ ( |s “ ( ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∩ <<s ) ) ↔ ∃ 𝑦 ∈ ( ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∩ <<s ) 𝑦 |s 𝑥 ) |
22 |
|
elin |
⊢ ( 𝑦 ∈ ( ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∩ <<s ) ↔ ( 𝑦 ∈ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∧ 𝑦 ∈ <<s ) ) |
23 |
22
|
anbi1i |
⊢ ( ( 𝑦 ∈ ( ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∩ <<s ) ∧ 𝑦 |s 𝑥 ) ↔ ( ( 𝑦 ∈ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∧ 𝑦 ∈ <<s ) ∧ 𝑦 |s 𝑥 ) ) |
24 |
|
anass |
⊢ ( ( ( 𝑦 ∈ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∧ 𝑦 ∈ <<s ) ∧ 𝑦 |s 𝑥 ) ↔ ( 𝑦 ∈ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∧ ( 𝑦 ∈ <<s ∧ 𝑦 |s 𝑥 ) ) ) |
25 |
23 24
|
bitri |
⊢ ( ( 𝑦 ∈ ( ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∩ <<s ) ∧ 𝑦 |s 𝑥 ) ↔ ( 𝑦 ∈ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∧ ( 𝑦 ∈ <<s ∧ 𝑦 |s 𝑥 ) ) ) |
26 |
25
|
rexbii2 |
⊢ ( ∃ 𝑦 ∈ ( ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ∩ <<s ) 𝑦 |s 𝑥 ↔ ∃ 𝑦 ∈ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ( 𝑦 ∈ <<s ∧ 𝑦 |s 𝑥 ) ) |
27 |
19 21 26
|
3bitri |
⊢ ( 𝑥 ∈ ( |s “ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ) ↔ ∃ 𝑦 ∈ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ( 𝑦 ∈ <<s ∧ 𝑦 |s 𝑥 ) ) |
28 |
|
df-br |
⊢ ( 𝑎 <<s 𝑏 ↔ 〈 𝑎 , 𝑏 〉 ∈ <<s ) |
29 |
28
|
anbi1i |
⊢ ( ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ↔ ( 〈 𝑎 , 𝑏 〉 ∈ <<s ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ) |
30 |
|
df-ov |
⊢ ( 𝑎 |s 𝑏 ) = ( |s ‘ 〈 𝑎 , 𝑏 〉 ) |
31 |
30
|
eqeq1i |
⊢ ( ( 𝑎 |s 𝑏 ) = 𝑥 ↔ ( |s ‘ 〈 𝑎 , 𝑏 〉 ) = 𝑥 ) |
32 |
|
scutf |
⊢ |s : <<s ⟶ No |
33 |
|
ffn |
⊢ ( |s : <<s ⟶ No → |s Fn <<s ) |
34 |
32 33
|
ax-mp |
⊢ |s Fn <<s |
35 |
|
fnbrfvb |
⊢ ( ( |s Fn <<s ∧ 〈 𝑎 , 𝑏 〉 ∈ <<s ) → ( ( |s ‘ 〈 𝑎 , 𝑏 〉 ) = 𝑥 ↔ 〈 𝑎 , 𝑏 〉 |s 𝑥 ) ) |
36 |
34 35
|
mpan |
⊢ ( 〈 𝑎 , 𝑏 〉 ∈ <<s → ( ( |s ‘ 〈 𝑎 , 𝑏 〉 ) = 𝑥 ↔ 〈 𝑎 , 𝑏 〉 |s 𝑥 ) ) |
37 |
31 36
|
syl5bb |
⊢ ( 〈 𝑎 , 𝑏 〉 ∈ <<s → ( ( 𝑎 |s 𝑏 ) = 𝑥 ↔ 〈 𝑎 , 𝑏 〉 |s 𝑥 ) ) |
38 |
37
|
pm5.32i |
⊢ ( ( 〈 𝑎 , 𝑏 〉 ∈ <<s ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ↔ ( 〈 𝑎 , 𝑏 〉 ∈ <<s ∧ 〈 𝑎 , 𝑏 〉 |s 𝑥 ) ) |
39 |
29 38
|
bitri |
⊢ ( ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ↔ ( 〈 𝑎 , 𝑏 〉 ∈ <<s ∧ 〈 𝑎 , 𝑏 〉 |s 𝑥 ) ) |
40 |
39
|
2rexbii |
⊢ ( ∃ 𝑎 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ↔ ∃ 𝑎 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 〈 𝑎 , 𝑏 〉 ∈ <<s ∧ 〈 𝑎 , 𝑏 〉 |s 𝑥 ) ) |
41 |
13 27 40
|
3bitr4i |
⊢ ( 𝑥 ∈ ( |s “ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ) ↔ ∃ 𝑎 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ) |
42 |
41
|
abbi2i |
⊢ ( |s “ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ) = { 𝑥 ∣ ∃ 𝑎 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) } |
43 |
|
df-rab |
⊢ { 𝑥 ∈ No ∣ ∃ 𝑎 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) } = { 𝑥 ∣ ( 𝑥 ∈ No ∧ ∃ 𝑎 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) ) } |
44 |
9 42 43
|
3eqtr4i |
⊢ ( |s “ ( 𝒫 ∪ ( M “ 𝐴 ) × 𝒫 ∪ ( M “ 𝐴 ) ) ) = { 𝑥 ∈ No ∣ ∃ 𝑎 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) } |
45 |
1 44
|
eqtrdi |
⊢ ( 𝐴 ∈ On → ( M ‘ 𝐴 ) = { 𝑥 ∈ No ∣ ∃ 𝑎 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑏 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑎 <<s 𝑏 ∧ ( 𝑎 |s 𝑏 ) = 𝑥 ) } ) |