Definition df-pstm 29932

Description: Define the metric induced by a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)

Assertion

Ref	Expression
df-pstm	|- pstoMet = ( d e. U. ran PsMet |-> ( a e. ( dom dom d /. (~Met ` d ) ) , b e. ( dom dom d /. (~Met ` d ) ) |-> U. { z | E. x e. a E. y e. b z = ( x d y ) } ) )

Distinct variable group:   a, b, d, x, y, z

Detailed syntax breakdown of Definition df-pstm

Step	Hyp	Ref	Expression
1		cpstm 29930	. 2 class pstoMet
2		vd	. . 3 setvar d
3		cpsmet 19730	. . . . 5 class PsMet
4	3	crn 5115	. . . 4 class ran PsMet
5	4	cuni 4436	. . 3 class U. ran PsMet
6		va	. . . 4 setvar a
7		vb	. . . 4 setvar b
8	2	cv 1482	. . . . . . 7 class d
9	8	cdm 5114	. . . . . 6 class dom d
10	9	cdm 5114	. . . . 5 class dom dom d
11		cmetid 29929	. . . . . 6 class ~Met
12	8, 11	cfv 5888	. . . . 5 class (~Met ` d )
13	10, 12	cqs 7741	. . . 4 class ( dom dom d /. (~Met ` d ) )
14		vz	. . . . . . . . . 10 setvar z
15	14	cv 1482	. . . . . . . . 9 class z
16		vx	. . . . . . . . . . 11 setvar x
17	16	cv 1482	. . . . . . . . . 10 class x
18		vy	. . . . . . . . . . 11 setvar y
19	18	cv 1482	. . . . . . . . . 10 class y
20	17, 19, 8	co 6650	. . . . . . . . 9 class ( x d y )
21	15, 20	wceq 1483	. . . . . . . 8 wff z = ( x d y )
22	7	cv 1482	. . . . . . . 8 class b
23	21, 18, 22	wrex 2913	. . . . . . 7 wff E. y e. b z = ( x d y )
24	6	cv 1482	. . . . . . 7 class a
25	23, 16, 24	wrex 2913	. . . . . 6 wff E. x e. a E. y e. b z = ( x d y )
26	25, 14	cab 2608	. . . . 5 class { z | E. x e. a E. y e. b z = ( x d y ) }
27	26	cuni 4436	. . . 4 class U. { z | E. x e. a E. y e. b z = ( x d y ) }
28	6, 7, 13, 13, 27	cmpt2 6652	. . 3 class ( a e. ( dom dom d /. (~Met ` d ) ) , b e. ( dom dom d /. (~Met ` d ) ) |-> U. { z | E. x e. a E. y e. b z = ( x d y ) } )
29	2, 5, 28	cmpt 4729	. 2 class ( d e. U. ran PsMet |-> ( a e. ( dom dom d /. (~Met ` d ) ) , b e. ( dom dom d /. (~Met ` d ) ) |-> U. { z | E. x e. a E. y e. b z = ( x d y ) } ) )
30	1, 29	wceq 1483	1 wff pstoMet = ( d e. U. ran PsMet |-> ( a e. ( dom dom d /. (~Met ` d ) ) , b e. ( dom dom d /. (~Met ` d ) ) |-> U. { z | E. x e. a E. y e. b z = ( x d y ) } ) )

This definition is referenced by:  pstmval  29938