Step |
Hyp |
Ref |
Expression |
0 |
|
cpconn |
⊢ PConn |
1 |
|
vj |
⊢ 𝑗 |
2 |
|
ctop |
⊢ Top |
3 |
|
vx |
⊢ 𝑥 |
4 |
1
|
cv |
⊢ 𝑗 |
5 |
4
|
cuni |
⊢ ∪ 𝑗 |
6 |
|
vy |
⊢ 𝑦 |
7 |
|
vf |
⊢ 𝑓 |
8 |
|
cii |
⊢ II |
9 |
|
ccn |
⊢ Cn |
10 |
8 4 9
|
co |
⊢ ( II Cn 𝑗 ) |
11 |
7
|
cv |
⊢ 𝑓 |
12 |
|
cc0 |
⊢ 0 |
13 |
12 11
|
cfv |
⊢ ( 𝑓 ‘ 0 ) |
14 |
3
|
cv |
⊢ 𝑥 |
15 |
13 14
|
wceq |
⊢ ( 𝑓 ‘ 0 ) = 𝑥 |
16 |
|
c1 |
⊢ 1 |
17 |
16 11
|
cfv |
⊢ ( 𝑓 ‘ 1 ) |
18 |
6
|
cv |
⊢ 𝑦 |
19 |
17 18
|
wceq |
⊢ ( 𝑓 ‘ 1 ) = 𝑦 |
20 |
15 19
|
wa |
⊢ ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) |
21 |
20 7 10
|
wrex |
⊢ ∃ 𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) |
22 |
21 6 5
|
wral |
⊢ ∀ 𝑦 ∈ ∪ 𝑗 ∃ 𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) |
23 |
22 3 5
|
wral |
⊢ ∀ 𝑥 ∈ ∪ 𝑗 ∀ 𝑦 ∈ ∪ 𝑗 ∃ 𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) |
24 |
23 1 2
|
crab |
⊢ { 𝑗 ∈ Top ∣ ∀ 𝑥 ∈ ∪ 𝑗 ∀ 𝑦 ∈ ∪ 𝑗 ∃ 𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } |
25 |
0 24
|
wceq |
⊢ PConn = { 𝑗 ∈ Top ∣ ∀ 𝑥 ∈ ∪ 𝑗 ∀ 𝑦 ∈ ∪ 𝑗 ∃ 𝑓 ∈ ( II Cn 𝑗 ) ( ( 𝑓 ‘ 0 ) = 𝑥 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } |