Definition df-usp 22061

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 ) e. (UnifOn ` ( Base ` f ) ) /\ ( TopOpen ` f ) = (unifTop ` (UnifSt ` f ) ) ) }

Detailed syntax breakdown of Definition df-usp

Step	Hyp	Ref	Expression
1		cusp 22058	. 2 class UnifSp
2		vf	. . . . . . 7 setvar f
3	2	cv 1482	. . . . . 6 class f
4		cuss 22057	. . . . . 6 class UnifSt
5	3, 4	cfv 5888	. . . . 5 class (UnifSt ` f )
6		cbs 15857	. . . . . . 7 class Base
7	3, 6	cfv 5888	. . . . . 6 class ( Base ` f )
8		cust 22003	. . . . . 6 class UnifOn
9	7, 8	cfv 5888	. . . . 5 class (UnifOn ` ( Base ` f ) )
10	5, 9	wcel 1990	. . . 4 wff (UnifSt ` f ) e. (UnifOn ` ( Base ` f ) )
11		ctopn 16082	. . . . . 6 class TopOpen
12	3, 11	cfv 5888	. . . . 5 class ( TopOpen ` f )
13		cutop 22034	. . . . . 6 class unifTop
14	5, 13	cfv 5888	. . . . 5 class (unifTop ` (UnifSt ` f ) )
15	12, 14	wceq 1483	. . . 4 wff ( TopOpen ` f ) = (unifTop ` (UnifSt ` f ) )
16	10, 15	wa 384	. . 3 wff ( (UnifSt ` f ) e. (UnifOn ` ( Base ` f ) ) /\ ( TopOpen ` f ) = (unifTop ` (UnifSt ` f ) ) )
17	16, 2	cab 2608	. 2 class { f | ( (UnifSt ` f ) e. (UnifOn ` ( Base ` f ) ) /\ ( TopOpen ` f ) = (unifTop ` (UnifSt ` f ) ) ) }
18	1, 17	wceq 1483	1 wff UnifSp = { f | ( (UnifSt ` f ) e. (UnifOn ` ( Base ` f ) ) /\ ( TopOpen ` f ) = (unifTop ` (UnifSt ` f ) ) ) }

This definition is referenced by:  isusp  22065