Step |
Hyp |
Ref |
Expression |
0 |
|
cnadd |
⊢ +no |
1 |
|
vx |
⊢ 𝑥 |
2 |
|
vy |
⊢ 𝑦 |
3 |
1
|
cv |
⊢ 𝑥 |
4 |
|
con0 |
⊢ On |
5 |
4 4
|
cxp |
⊢ ( On × On ) |
6 |
3 5
|
wcel |
⊢ 𝑥 ∈ ( On × On ) |
7 |
2
|
cv |
⊢ 𝑦 |
8 |
7 5
|
wcel |
⊢ 𝑦 ∈ ( On × On ) |
9 |
|
c1st |
⊢ 1st |
10 |
3 9
|
cfv |
⊢ ( 1st ‘ 𝑥 ) |
11 |
|
cep |
⊢ E |
12 |
7 9
|
cfv |
⊢ ( 1st ‘ 𝑦 ) |
13 |
10 12 11
|
wbr |
⊢ ( 1st ‘ 𝑥 ) E ( 1st ‘ 𝑦 ) |
14 |
10 12
|
wceq |
⊢ ( 1st ‘ 𝑥 ) = ( 1st ‘ 𝑦 ) |
15 |
13 14
|
wo |
⊢ ( ( 1st ‘ 𝑥 ) E ( 1st ‘ 𝑦 ) ∨ ( 1st ‘ 𝑥 ) = ( 1st ‘ 𝑦 ) ) |
16 |
|
c2nd |
⊢ 2nd |
17 |
3 16
|
cfv |
⊢ ( 2nd ‘ 𝑥 ) |
18 |
7 16
|
cfv |
⊢ ( 2nd ‘ 𝑦 ) |
19 |
17 18 11
|
wbr |
⊢ ( 2nd ‘ 𝑥 ) E ( 2nd ‘ 𝑦 ) |
20 |
17 18
|
wceq |
⊢ ( 2nd ‘ 𝑥 ) = ( 2nd ‘ 𝑦 ) |
21 |
19 20
|
wo |
⊢ ( ( 2nd ‘ 𝑥 ) E ( 2nd ‘ 𝑦 ) ∨ ( 2nd ‘ 𝑥 ) = ( 2nd ‘ 𝑦 ) ) |
22 |
3 7
|
wne |
⊢ 𝑥 ≠ 𝑦 |
23 |
15 21 22
|
w3a |
⊢ ( ( ( 1st ‘ 𝑥 ) E ( 1st ‘ 𝑦 ) ∨ ( 1st ‘ 𝑥 ) = ( 1st ‘ 𝑦 ) ) ∧ ( ( 2nd ‘ 𝑥 ) E ( 2nd ‘ 𝑦 ) ∨ ( 2nd ‘ 𝑥 ) = ( 2nd ‘ 𝑦 ) ) ∧ 𝑥 ≠ 𝑦 ) |
24 |
6 8 23
|
w3a |
⊢ ( 𝑥 ∈ ( On × On ) ∧ 𝑦 ∈ ( On × On ) ∧ ( ( ( 1st ‘ 𝑥 ) E ( 1st ‘ 𝑦 ) ∨ ( 1st ‘ 𝑥 ) = ( 1st ‘ 𝑦 ) ) ∧ ( ( 2nd ‘ 𝑥 ) E ( 2nd ‘ 𝑦 ) ∨ ( 2nd ‘ 𝑥 ) = ( 2nd ‘ 𝑦 ) ) ∧ 𝑥 ≠ 𝑦 ) ) |
25 |
24 1 2
|
copab |
⊢ { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ ( On × On ) ∧ 𝑦 ∈ ( On × On ) ∧ ( ( ( 1st ‘ 𝑥 ) E ( 1st ‘ 𝑦 ) ∨ ( 1st ‘ 𝑥 ) = ( 1st ‘ 𝑦 ) ) ∧ ( ( 2nd ‘ 𝑥 ) E ( 2nd ‘ 𝑦 ) ∨ ( 2nd ‘ 𝑥 ) = ( 2nd ‘ 𝑦 ) ) ∧ 𝑥 ≠ 𝑦 ) ) } |
26 |
|
vz |
⊢ 𝑧 |
27 |
|
cvv |
⊢ V |
28 |
|
va |
⊢ 𝑎 |
29 |
|
vw |
⊢ 𝑤 |
30 |
28
|
cv |
⊢ 𝑎 |
31 |
26
|
cv |
⊢ 𝑧 |
32 |
31 9
|
cfv |
⊢ ( 1st ‘ 𝑧 ) |
33 |
32
|
csn |
⊢ { ( 1st ‘ 𝑧 ) } |
34 |
31 16
|
cfv |
⊢ ( 2nd ‘ 𝑧 ) |
35 |
33 34
|
cxp |
⊢ ( { ( 1st ‘ 𝑧 ) } × ( 2nd ‘ 𝑧 ) ) |
36 |
30 35
|
cima |
⊢ ( 𝑎 “ ( { ( 1st ‘ 𝑧 ) } × ( 2nd ‘ 𝑧 ) ) ) |
37 |
29
|
cv |
⊢ 𝑤 |
38 |
36 37
|
wss |
⊢ ( 𝑎 “ ( { ( 1st ‘ 𝑧 ) } × ( 2nd ‘ 𝑧 ) ) ) ⊆ 𝑤 |
39 |
34
|
csn |
⊢ { ( 2nd ‘ 𝑧 ) } |
40 |
32 39
|
cxp |
⊢ ( ( 1st ‘ 𝑧 ) × { ( 2nd ‘ 𝑧 ) } ) |
41 |
30 40
|
cima |
⊢ ( 𝑎 “ ( ( 1st ‘ 𝑧 ) × { ( 2nd ‘ 𝑧 ) } ) ) |
42 |
41 37
|
wss |
⊢ ( 𝑎 “ ( ( 1st ‘ 𝑧 ) × { ( 2nd ‘ 𝑧 ) } ) ) ⊆ 𝑤 |
43 |
38 42
|
wa |
⊢ ( ( 𝑎 “ ( { ( 1st ‘ 𝑧 ) } × ( 2nd ‘ 𝑧 ) ) ) ⊆ 𝑤 ∧ ( 𝑎 “ ( ( 1st ‘ 𝑧 ) × { ( 2nd ‘ 𝑧 ) } ) ) ⊆ 𝑤 ) |
44 |
43 29 4
|
crab |
⊢ { 𝑤 ∈ On ∣ ( ( 𝑎 “ ( { ( 1st ‘ 𝑧 ) } × ( 2nd ‘ 𝑧 ) ) ) ⊆ 𝑤 ∧ ( 𝑎 “ ( ( 1st ‘ 𝑧 ) × { ( 2nd ‘ 𝑧 ) } ) ) ⊆ 𝑤 ) } |
45 |
44
|
cint |
⊢ ∩ { 𝑤 ∈ On ∣ ( ( 𝑎 “ ( { ( 1st ‘ 𝑧 ) } × ( 2nd ‘ 𝑧 ) ) ) ⊆ 𝑤 ∧ ( 𝑎 “ ( ( 1st ‘ 𝑧 ) × { ( 2nd ‘ 𝑧 ) } ) ) ⊆ 𝑤 ) } |
46 |
26 28 27 27 45
|
cmpo |
⊢ ( 𝑧 ∈ V , 𝑎 ∈ V ↦ ∩ { 𝑤 ∈ On ∣ ( ( 𝑎 “ ( { ( 1st ‘ 𝑧 ) } × ( 2nd ‘ 𝑧 ) ) ) ⊆ 𝑤 ∧ ( 𝑎 “ ( ( 1st ‘ 𝑧 ) × { ( 2nd ‘ 𝑧 ) } ) ) ⊆ 𝑤 ) } ) |
47 |
5 25 46
|
cfrecs |
⊢ frecs ( { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ ( On × On ) ∧ 𝑦 ∈ ( On × On ) ∧ ( ( ( 1st ‘ 𝑥 ) E ( 1st ‘ 𝑦 ) ∨ ( 1st ‘ 𝑥 ) = ( 1st ‘ 𝑦 ) ) ∧ ( ( 2nd ‘ 𝑥 ) E ( 2nd ‘ 𝑦 ) ∨ ( 2nd ‘ 𝑥 ) = ( 2nd ‘ 𝑦 ) ) ∧ 𝑥 ≠ 𝑦 ) ) } , ( On × On ) , ( 𝑧 ∈ V , 𝑎 ∈ V ↦ ∩ { 𝑤 ∈ On ∣ ( ( 𝑎 “ ( { ( 1st ‘ 𝑧 ) } × ( 2nd ‘ 𝑧 ) ) ) ⊆ 𝑤 ∧ ( 𝑎 “ ( ( 1st ‘ 𝑧 ) × { ( 2nd ‘ 𝑧 ) } ) ) ⊆ 𝑤 ) } ) ) |
48 |
0 47
|
wceq |
⊢ +no = frecs ( { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ ( On × On ) ∧ 𝑦 ∈ ( On × On ) ∧ ( ( ( 1st ‘ 𝑥 ) E ( 1st ‘ 𝑦 ) ∨ ( 1st ‘ 𝑥 ) = ( 1st ‘ 𝑦 ) ) ∧ ( ( 2nd ‘ 𝑥 ) E ( 2nd ‘ 𝑦 ) ∨ ( 2nd ‘ 𝑥 ) = ( 2nd ‘ 𝑦 ) ) ∧ 𝑥 ≠ 𝑦 ) ) } , ( On × On ) , ( 𝑧 ∈ V , 𝑎 ∈ V ↦ ∩ { 𝑤 ∈ On ∣ ( ( 𝑎 “ ( { ( 1st ‘ 𝑧 ) } × ( 2nd ‘ 𝑧 ) ) ) ⊆ 𝑤 ∧ ( 𝑎 “ ( ( 1st ‘ 𝑧 ) × { ( 2nd ‘ 𝑧 ) } ) ) ⊆ 𝑤 ) } ) ) |