Detailed syntax breakdown of Definition df-preset
Step | Hyp | Ref
| Expression |
1 | | cpreset 16926 |
. 2
class
Preset |
2 | | vx |
. . . . . . . . . . 11
setvar 𝑥 |
3 | 2 | cv 1482 |
. . . . . . . . . 10
class 𝑥 |
4 | | vr |
. . . . . . . . . . 11
setvar 𝑟 |
5 | 4 | cv 1482 |
. . . . . . . . . 10
class 𝑟 |
6 | 3, 3, 5 | wbr 4653 |
. . . . . . . . 9
wff 𝑥𝑟𝑥 |
7 | | vy |
. . . . . . . . . . . . 13
setvar 𝑦 |
8 | 7 | cv 1482 |
. . . . . . . . . . . 12
class 𝑦 |
9 | 3, 8, 5 | wbr 4653 |
. . . . . . . . . . 11
wff 𝑥𝑟𝑦 |
10 | | vz |
. . . . . . . . . . . . 13
setvar 𝑧 |
11 | 10 | cv 1482 |
. . . . . . . . . . . 12
class 𝑧 |
12 | 8, 11, 5 | wbr 4653 |
. . . . . . . . . . 11
wff 𝑦𝑟𝑧 |
13 | 9, 12 | wa 384 |
. . . . . . . . . 10
wff (𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) |
14 | 3, 11, 5 | wbr 4653 |
. . . . . . . . . 10
wff 𝑥𝑟𝑧 |
15 | 13, 14 | wi 4 |
. . . . . . . . 9
wff ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧) |
16 | 6, 15 | wa 384 |
. . . . . . . 8
wff (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) |
17 | | vb |
. . . . . . . . 9
setvar 𝑏 |
18 | 17 | cv 1482 |
. . . . . . . 8
class 𝑏 |
19 | 16, 10, 18 | wral 2912 |
. . . . . . 7
wff
∀𝑧 ∈
𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) |
20 | 19, 7, 18 | wral 2912 |
. . . . . 6
wff
∀𝑦 ∈
𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) |
21 | 20, 2, 18 | wral 2912 |
. . . . 5
wff
∀𝑥 ∈
𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) |
22 | | vf |
. . . . . . 7
setvar 𝑓 |
23 | 22 | cv 1482 |
. . . . . 6
class 𝑓 |
24 | | cple 15948 |
. . . . . 6
class
le |
25 | 23, 24 | cfv 5888 |
. . . . 5
class
(le‘𝑓) |
26 | 21, 4, 25 | wsbc 3435 |
. . . 4
wff
[(le‘𝑓)
/ 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) |
27 | | cbs 15857 |
. . . . 5
class
Base |
28 | 23, 27 | cfv 5888 |
. . . 4
class
(Base‘𝑓) |
29 | 26, 17, 28 | wsbc 3435 |
. . 3
wff
[(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)) |
30 | 29, 22 | cab 2608 |
. 2
class {𝑓 ∣
[(Base‘𝑓) /
𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))} |
31 | 1, 30 | wceq 1483 |
1
wff Preset =
{𝑓 ∣
[(Base‘𝑓) /
𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))} |