Detailed syntax breakdown of Definition df-xms
Step | Hyp | Ref
| Expression |
1 | | cxms 23479 |
. 2
class
∞MetSp |
2 | | vf |
. . . . . 6
setvar 𝑓 |
3 | 2 | cv 1538 |
. . . . 5
class 𝑓 |
4 | | ctopn 17141 |
. . . . 5
class
TopOpen |
5 | 3, 4 | cfv 6437 |
. . . 4
class
(TopOpen‘𝑓) |
6 | | cds 16980 |
. . . . . . 7
class
dist |
7 | 3, 6 | cfv 6437 |
. . . . . 6
class
(dist‘𝑓) |
8 | | cbs 16921 |
. . . . . . . 8
class
Base |
9 | 3, 8 | cfv 6437 |
. . . . . . 7
class
(Base‘𝑓) |
10 | 9, 9 | cxp 5588 |
. . . . . 6
class
((Base‘𝑓)
× (Base‘𝑓)) |
11 | 7, 10 | cres 5592 |
. . . . 5
class
((dist‘𝑓)
↾ ((Base‘𝑓)
× (Base‘𝑓))) |
12 | | cmopn 20596 |
. . . . 5
class
MetOpen |
13 | 11, 12 | cfv 6437 |
. . . 4
class
(MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) |
14 | 5, 13 | wceq 1539 |
. . 3
wff
(TopOpen‘𝑓) =
(MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) |
15 | | ctps 22090 |
. . 3
class
TopSp |
16 | 14, 2, 15 | crab 3069 |
. 2
class {𝑓 ∈ TopSp ∣
(TopOpen‘𝑓) =
(MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))} |
17 | 1, 16 | wceq 1539 |
1
wff
∞MetSp = {𝑓
∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))} |