Step |
Hyp |
Ref |
Expression |
0 |
|
comi |
⊢ Ω1 |
1 |
|
vj |
⊢ 𝑗 |
2 |
|
ctop |
⊢ Top |
3 |
|
vy |
⊢ 𝑦 |
4 |
1
|
cv |
⊢ 𝑗 |
5 |
4
|
cuni |
⊢ ∪ 𝑗 |
6 |
|
cbs |
⊢ Base |
7 |
|
cnx |
⊢ ndx |
8 |
7 6
|
cfv |
⊢ ( Base ‘ ndx ) |
9 |
|
vf |
⊢ 𝑓 |
10 |
|
cii |
⊢ II |
11 |
|
ccn |
⊢ Cn |
12 |
10 4 11
|
co |
⊢ ( II Cn 𝑗 ) |
13 |
9
|
cv |
⊢ 𝑓 |
14 |
|
cc0 |
⊢ 0 |
15 |
14 13
|
cfv |
⊢ ( 𝑓 ‘ 0 ) |
16 |
3
|
cv |
⊢ 𝑦 |
17 |
15 16
|
wceq |
⊢ ( 𝑓 ‘ 0 ) = 𝑦 |
18 |
|
c1 |
⊢ 1 |
19 |
18 13
|
cfv |
⊢ ( 𝑓 ‘ 1 ) |
20 |
19 16
|
wceq |
⊢ ( 𝑓 ‘ 1 ) = 𝑦 |
21 |
17 20
|
wa |
⊢ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) |
22 |
21 9 12
|
crab |
⊢ { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } |
23 |
8 22
|
cop |
⊢ 〈 ( Base ‘ ndx ) , { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } 〉 |
24 |
|
cplusg |
⊢ +g |
25 |
7 24
|
cfv |
⊢ ( +g ‘ ndx ) |
26 |
|
cpco |
⊢ *𝑝 |
27 |
4 26
|
cfv |
⊢ ( *𝑝 ‘ 𝑗 ) |
28 |
25 27
|
cop |
⊢ 〈 ( +g ‘ ndx ) , ( *𝑝 ‘ 𝑗 ) 〉 |
29 |
|
cts |
⊢ TopSet |
30 |
7 29
|
cfv |
⊢ ( TopSet ‘ ndx ) |
31 |
|
cxko |
⊢ ↑ko |
32 |
4 10 31
|
co |
⊢ ( 𝑗 ↑ko II ) |
33 |
30 32
|
cop |
⊢ 〈 ( TopSet ‘ ndx ) , ( 𝑗 ↑ko II ) 〉 |
34 |
23 28 33
|
ctp |
⊢ { 〈 ( Base ‘ ndx ) , { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } 〉 , 〈 ( +g ‘ ndx ) , ( *𝑝 ‘ 𝑗 ) 〉 , 〈 ( TopSet ‘ ndx ) , ( 𝑗 ↑ko II ) 〉 } |
35 |
1 3 2 5 34
|
cmpo |
⊢ ( 𝑗 ∈ Top , 𝑦 ∈ ∪ 𝑗 ↦ { 〈 ( Base ‘ ndx ) , { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } 〉 , 〈 ( +g ‘ ndx ) , ( *𝑝 ‘ 𝑗 ) 〉 , 〈 ( TopSet ‘ ndx ) , ( 𝑗 ↑ko II ) 〉 } ) |
36 |
0 35
|
wceq |
⊢ Ω1 = ( 𝑗 ∈ Top , 𝑦 ∈ ∪ 𝑗 ↦ { 〈 ( Base ‘ ndx ) , { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } 〉 , 〈 ( +g ‘ ndx ) , ( *𝑝 ‘ 𝑗 ) 〉 , 〈 ( TopSet ‘ ndx ) , ( 𝑗 ↑ko II ) 〉 } ) |