Definition df-pstm 33888

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

Assertion

Ref	Expression
df-pstm	⊢ pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}))

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

Detailed syntax breakdown of Definition df-pstm

Step	Hyp	Ref	Expression
1		cpstm 33886	. 2 class pstoMet
2		vd	. . 3 setvar 𝑑
3		cpsmet 21348	. . . . 5 class PsMet
4	3	crn 5686	. . . 4 class ran PsMet
5	4	cuni 4907	. . 3 class ∪ ran PsMet
6		va	. . . 4 setvar 𝑎
7		vb	. . . 4 setvar 𝑏
8	2	cv 1539	. . . . . . 7 class 𝑑
9	8	cdm 5685	. . . . . 6 class dom 𝑑
10	9	cdm 5685	. . . . 5 class dom dom 𝑑
11		cmetid 33885	. . . . . 6 class ~Met
12	8, 11	cfv 6561	. . . . 5 class (~Met‘𝑑)
13	10, 12	cqs 8744	. . . 4 class (dom dom 𝑑 / (~Met‘𝑑))
14		vz	. . . . . . . . . 10 setvar 𝑧
15	14	cv 1539	. . . . . . . . 9 class 𝑧
16		vx	. . . . . . . . . . 11 setvar 𝑥
17	16	cv 1539	. . . . . . . . . 10 class 𝑥
18		vy	. . . . . . . . . . 11 setvar 𝑦
19	18	cv 1539	. . . . . . . . . 10 class 𝑦
20	17, 19, 8	co 7431	. . . . . . . . 9 class (𝑥𝑑𝑦)
21	15, 20	wceq 1540	. . . . . . . 8 wff 𝑧 = (𝑥𝑑𝑦)
22	7	cv 1539	. . . . . . . 8 class 𝑏
23	21, 18, 22	wrex 3070	. . . . . . 7 wff ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)
24	6	cv 1539	. . . . . . 7 class 𝑎
25	23, 16, 24	wrex 3070	. . . . . 6 wff ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)
26	25, 14	cab 2714	. . . . 5 class {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}
27	26	cuni 4907	. . . 4 class ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}
28	6, 7, 13, 13, 27	cmpo 7433	. . 3 class (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)})
29	2, 5, 28	cmpt 5225	. 2 class (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}))
30	1, 29	wceq 1540	1 wff pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}))

This definition is referenced by:  pstmval  33894