Definition df-usp 23317

Description: Definition of a uniform space, i.e. a base set with an uniform structure and its induced topology. Derived from definition 3 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Nov-2017.)

Assertion

Ref	Expression
df-usp	⊢ UnifSp = {𝑓 ∣ ((UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓)) ∧ (TopOpen‘𝑓) = (unifTop‘(UnifSt‘𝑓)))}

Detailed syntax breakdown of Definition df-usp

Step	Hyp	Ref	Expression
1		cusp 23314	. 2 class UnifSp
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1538	. . . . . 6 class 𝑓
4		cuss 23313	. . . . . 6 class UnifSt
5	3, 4	cfv 6418	. . . . 5 class (UnifSt‘𝑓)
6		cbs 16840	. . . . . . 7 class Base
7	3, 6	cfv 6418	. . . . . 6 class (Base‘𝑓)
8		cust 23259	. . . . . 6 class UnifOn
9	7, 8	cfv 6418	. . . . 5 class (UnifOn‘(Base‘𝑓))
10	5, 9	wcel 2108	. . . 4 wff (UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓))
11		ctopn 17049	. . . . . 6 class TopOpen
12	3, 11	cfv 6418	. . . . 5 class (TopOpen‘𝑓)
13		cutop 23290	. . . . . 6 class unifTop
14	5, 13	cfv 6418	. . . . 5 class (unifTop‘(UnifSt‘𝑓))
15	12, 14	wceq 1539	. . . 4 wff (TopOpen‘𝑓) = (unifTop‘(UnifSt‘𝑓))
16	10, 15	wa 395	. . 3 wff ((UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓)) ∧ (TopOpen‘𝑓) = (unifTop‘(UnifSt‘𝑓)))
17	16, 2	cab 2715	. 2 class {𝑓 ∣ ((UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓)) ∧ (TopOpen‘𝑓) = (unifTop‘(UnifSt‘𝑓)))}
18	1, 17	wceq 1539	1 wff UnifSp = {𝑓 ∣ ((UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓)) ∧ (TopOpen‘𝑓) = (unifTop‘(UnifSt‘𝑓)))}

This definition is referenced by:  isusp  23321