Step |
Hyp |
Ref |
Expression |
0 |
|
ctos |
⊢ Toset |
1 |
|
vf |
⊢ 𝑓 |
2 |
|
cpo |
⊢ Poset |
3 |
|
cbs |
⊢ Base |
4 |
1
|
cv |
⊢ 𝑓 |
5 |
4 3
|
cfv |
⊢ ( Base ‘ 𝑓 ) |
6 |
|
vb |
⊢ 𝑏 |
7 |
|
cple |
⊢ le |
8 |
4 7
|
cfv |
⊢ ( le ‘ 𝑓 ) |
9 |
|
vr |
⊢ 𝑟 |
10 |
|
vx |
⊢ 𝑥 |
11 |
6
|
cv |
⊢ 𝑏 |
12 |
|
vy |
⊢ 𝑦 |
13 |
10
|
cv |
⊢ 𝑥 |
14 |
9
|
cv |
⊢ 𝑟 |
15 |
12
|
cv |
⊢ 𝑦 |
16 |
13 15 14
|
wbr |
⊢ 𝑥 𝑟 𝑦 |
17 |
15 13 14
|
wbr |
⊢ 𝑦 𝑟 𝑥 |
18 |
16 17
|
wo |
⊢ ( 𝑥 𝑟 𝑦 ∨ 𝑦 𝑟 𝑥 ) |
19 |
18 12 11
|
wral |
⊢ ∀ 𝑦 ∈ 𝑏 ( 𝑥 𝑟 𝑦 ∨ 𝑦 𝑟 𝑥 ) |
20 |
19 10 11
|
wral |
⊢ ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ( 𝑥 𝑟 𝑦 ∨ 𝑦 𝑟 𝑥 ) |
21 |
20 9 8
|
wsbc |
⊢ [ ( le ‘ 𝑓 ) / 𝑟 ] ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ( 𝑥 𝑟 𝑦 ∨ 𝑦 𝑟 𝑥 ) |
22 |
21 6 5
|
wsbc |
⊢ [ ( Base ‘ 𝑓 ) / 𝑏 ] [ ( le ‘ 𝑓 ) / 𝑟 ] ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ( 𝑥 𝑟 𝑦 ∨ 𝑦 𝑟 𝑥 ) |
23 |
22 1 2
|
crab |
⊢ { 𝑓 ∈ Poset ∣ [ ( Base ‘ 𝑓 ) / 𝑏 ] [ ( le ‘ 𝑓 ) / 𝑟 ] ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ( 𝑥 𝑟 𝑦 ∨ 𝑦 𝑟 𝑥 ) } |
24 |
0 23
|
wceq |
⊢ Toset = { 𝑓 ∈ Poset ∣ [ ( Base ‘ 𝑓 ) / 𝑏 ] [ ( le ‘ 𝑓 ) / 𝑟 ] ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ( 𝑥 𝑟 𝑦 ∨ 𝑦 𝑟 𝑥 ) } |