Theorem pstmfval 34414

Description: Function value of the metric induced by a pseudometric 𝐷 (Contributed by Thierry Arnoux, 11-Feb-2018.)

Hypothesis

Ref	Expression
pstmval.1	⊢ ∼ = (~Met‘𝐷)

Assertion

Ref	Expression
pstmfval	⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ([𝐴] ∼ (pstoMet‘𝐷)[𝐵] ∼ ) = (𝐴𝐷𝐵))

Proof of Theorem pstmfval

Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pstmval.1	. . . . 5 ⊢ ∼ = (~Met‘𝐷)
2	1	pstmval 34413	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}))
3	2	3ad2ant1 1151	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}))
4	3	oveqd 7434	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ([𝐴] ∼ (pstoMet‘𝐷)[𝐵] ∼ ) = ([𝐴] ∼ (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)})[𝐵] ∼ ))
5	1	fvexi 6896	. . . . 5 ⊢ ∼ ∈ V
6	5	ecelqsi 8773	. . . 4 ⊢ (𝐴 ∈ 𝑋 → [𝐴] ∼ ∈ (𝑋 / ∼ ))
7	6	3ad2ant2 1152	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → [𝐴] ∼ ∈ (𝑋 / ∼ ))
8	5	ecelqsi 8773	. . . 4 ⊢ (𝐵 ∈ 𝑋 → [𝐵] ∼ ∈ (𝑋 / ∼ ))
9	8	3ad2ant3 1153	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → [𝐵] ∼ ∈ (𝑋 / ∼ ))
10		rexeq 3317	. . . . . 6 ⊢ (𝑥 = [𝐴] ∼ → (∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)))
11	10	abbidv 2828	. . . . 5 ⊢ (𝑥 = [𝐴] ∼ → {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)})
12	11	unieqd 4883	. . . 4 ⊢ (𝑥 = [𝐴] ∼ → ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} = ∪ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)})
13		rexeq 3317	. . . . . . 7 ⊢ (𝑦 = [𝐵] ∼ → (∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)))
14	13	rexbidv 3188	. . . . . 6 ⊢ (𝑦 = [𝐵] ∼ → (∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)))
15	14	abbidv 2828	. . . . 5 ⊢ (𝑦 = [𝐵] ∼ → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)})
16	15	unieqd 4883	. . . 4 ⊢ (𝑦 = [𝐵] ∼ → ∪ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} = ∪ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)})
17		eqid 2762	. . . 4 ⊢ (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}) = (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)})
18		ecexg 8704	. . . . . . 7 ⊢ ( ∼ ∈ V → [𝐴] ∼ ∈ V)
19	5, 18	ax-mp 5	. . . . . 6 ⊢ [𝐴] ∼ ∈ V
20		ecexg 8704	. . . . . . 7 ⊢ ( ∼ ∈ V → [𝐵] ∼ ∈ V)
21	5, 20	ax-mp 5	. . . . . 6 ⊢ [𝐵] ∼ ∈ V
22	19, 21	ab2rexex 7980	. . . . 5 ⊢ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)} ∈ V
23	22	uniex 7747	. . . 4 ⊢ ∪ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)} ∈ V
24	12, 16, 17, 23	ovmpo 7577	. . 3 ⊢ (([𝐴] ∼ ∈ (𝑋 / ∼ ) ∧ [𝐵] ∼ ∈ (𝑋 / ∼ )) → ([𝐴] ∼ (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)})[𝐵] ∼ ) = ∪ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)})
25	7, 9, 24	syl2anc 596	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ([𝐴] ∼ (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)})[𝐵] ∼ ) = ∪ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)})
26		simpr3 1215	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝑒𝐷𝑓))
27		simpl1 1210	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝐷 ∈ (PsMet‘𝑋))
28		simpr1 1213	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝑒 ∈ [𝐴] ∼ )
29		metidss 34409	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) ⊆ (𝑋 × 𝑋))
30	1, 29	eqsstrid 3972	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∼ ⊆ (𝑋 × 𝑋))
31		xpss 5675	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 × 𝑋) ⊆ (V × V)
32	30, 31	sstrdi 3946	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∼ ⊆ (V × V))
33		df-rel 5666	. . . . . . . . . . . . . . . . . 18 ⊢ (Rel ∼ ↔ ∼ ⊆ (V × V))
34	32, 33	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ (PsMet‘𝑋) → Rel ∼ )
35	34	3ad2ant1 1151	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → Rel ∼ )
36	35	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → Rel ∼ )
37		relelec 8748	. . . . . . . . . . . . . . 15 ⊢ (Rel ∼ → (𝑒 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝑒))
38	36, 37	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → (𝑒 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝑒))
39	28, 38	mpbid 235	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝐴 ∼ 𝑒)
40	1	breqi 5113	. . . . . . . . . . . . 13 ⊢ (𝐴 ∼ 𝑒 ↔ 𝐴(~Met‘𝐷)𝑒)
41	39, 40	sylib 221	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝐴(~Met‘𝐷)𝑒)
42		simpr2 1214	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝑓 ∈ [𝐵] ∼ )
43		relelec 8748	. . . . . . . . . . . . . . 15 ⊢ (Rel ∼ → (𝑓 ∈ [𝐵] ∼ ↔ 𝐵 ∼ 𝑓))
44	36, 43	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → (𝑓 ∈ [𝐵] ∼ ↔ 𝐵 ∼ 𝑓))
45	42, 44	mpbid 235	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝐵 ∼ 𝑓)
46	1	breqi 5113	. . . . . . . . . . . . 13 ⊢ (𝐵 ∼ 𝑓 ↔ 𝐵(~Met‘𝐷)𝑓)
47	45, 46	sylib 221	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝐵(~Met‘𝐷)𝑓)
48		metideq 34411	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝑒 ∧ 𝐵(~Met‘𝐷)𝑓)) → (𝐴𝐷𝐵) = (𝑒𝐷𝑓))
49	27, 41, 47, 48	syl12anc 850	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → (𝐴𝐷𝐵) = (𝑒𝐷𝑓))
50	26, 49	eqtr4d 2800	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝐴𝐷𝐵))
51	50	adantlr 728	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝐴𝐷𝐵))
52	51	3anassrs 1381	. . . . . . . 8 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)) ∧ 𝑒 ∈ [𝐴] ∼ ) ∧ 𝑓 ∈ [𝐵] ∼ ) ∧ 𝑧 = (𝑒𝐷𝑓)) → 𝑧 = (𝐴𝐷𝐵))
53		oveq1 7424	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑒 → (𝑎𝐷𝑏) = (𝑒𝐷𝑏))
54	53	eqeq2d 2773	. . . . . . . . . 10 ⊢ (𝑎 = 𝑒 → (𝑧 = (𝑎𝐷𝑏) ↔ 𝑧 = (𝑒𝐷𝑏)))
55		oveq2 7425	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑓 → (𝑒𝐷𝑏) = (𝑒𝐷𝑓))
56	55	eqeq2d 2773	. . . . . . . . . 10 ⊢ (𝑏 = 𝑓 → (𝑧 = (𝑒𝐷𝑏) ↔ 𝑧 = (𝑒𝐷𝑓)))
57	54, 56	cbvrex2vw 3247	. . . . . . . . 9 ⊢ (∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑒 ∈ [ 𝐴] ∼ ∃𝑓 ∈ [ 𝐵] ∼ 𝑧 = (𝑒𝐷𝑓))
58	57	bilani 510	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)) → ∃𝑒 ∈ [ 𝐴] ∼ ∃𝑓 ∈ [ 𝐵] ∼ 𝑧 = (𝑒𝐷𝑓))
59	52, 58	r19.29vva 3224	. . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)) → 𝑧 = (𝐴𝐷𝐵))
60		simpl1 1210	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐷 ∈ (PsMet‘𝑋))
61		simpl2 1211	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐴 ∈ 𝑋)
62		psmet0 24540	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0)
63	60, 61, 62	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴𝐷𝐴) = 0)
64		relelec 8748	. . . . . . . . . . 11 ⊢ (Rel ∼ → (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
65	60, 34, 64	3syl 19	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
66	1	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → ∼ = (~Met‘𝐷))
67	66	breqd 5118	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 ∼ 𝐴 ↔ 𝐴(~Met‘𝐷)𝐴))
68		metidv 34410	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐴 ↔ (𝐴𝐷𝐴) = 0))
69	60, 61, 61, 68	syl12anc 850	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴(~Met‘𝐷)𝐴 ↔ (𝐴𝐷𝐴) = 0))
70	65, 67, 69	3bitrd 308	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 ∈ [𝐴] ∼ ↔ (𝐴𝐷𝐴) = 0))
71	63, 70	mpbird 260	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐴 ∈ [𝐴] ∼ )
72		simpl3 1212	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐵 ∈ 𝑋)
73		psmet0 24540	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 0)
74	60, 72, 73	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵𝐷𝐵) = 0)
75		relelec 8748	. . . . . . . . . . 11 ⊢ (Rel ∼ → (𝐵 ∈ [𝐵] ∼ ↔ 𝐵 ∼ 𝐵))
76	60, 34, 75	3syl 19	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 ∈ [𝐵] ∼ ↔ 𝐵 ∼ 𝐵))
77	66	breqd 5118	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 ∼ 𝐵 ↔ 𝐵(~Met‘𝐷)𝐵))
78		metidv 34410	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵(~Met‘𝐷)𝐵 ↔ (𝐵𝐷𝐵) = 0))
79	60, 72, 72, 78	syl12anc 850	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵(~Met‘𝐷)𝐵 ↔ (𝐵𝐷𝐵) = 0))
80	76, 77, 79	3bitrd 308	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 ∈ [𝐵] ∼ ↔ (𝐵𝐷𝐵) = 0))
81	74, 80	mpbird 260	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐵 ∈ [𝐵] ∼ )
82		simpr 490	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝑧 = (𝐴𝐷𝐵))
83		rspceov 7466	. . . . . . . 8 ⊢ ((𝐴 ∈ [𝐴] ∼ ∧ 𝐵 ∈ [𝐵] ∼ ∧ 𝑧 = (𝐴𝐷𝐵)) → ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏))
84	71, 81, 82, 83	syl3anc 1398	. . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏))
85	59, 84	impbida 813	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏) ↔ 𝑧 = (𝐴𝐷𝐵)))
86	85	abbidv 2828	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ 𝑧 = (𝐴𝐷𝐵)})
87		df-sn 4588	. . . . 5 ⊢ {(𝐴𝐷𝐵)} = {𝑧 ∣ 𝑧 = (𝐴𝐷𝐵)}
88	86, 87	eqtr4di 2815	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)} = {(𝐴𝐷𝐵)})
89	88	unieqd 4883	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∪ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)} = ∪ {(𝐴𝐷𝐵)})
90		ovex 7450	. . . 4 ⊢ (𝐴𝐷𝐵) ∈ V
91	90	unisn 4889	. . 3 ⊢ ∪ {(𝐴𝐷𝐵)} = (𝐴𝐷𝐵)
92	89, 91	eqtrdi 2813	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∪ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)} = (𝐴𝐷𝐵))
93	4, 25, 92	3eqtrd 2801	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ([𝐴] ∼ (pstoMet‘𝐷)[𝐵] ∼ ) = (𝐴𝐷𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2740  ∃wrex 3088  Vcvv 3453   ⊆ wss 3902  {csn 4587  ∪ cuni 4870   class class class wbr 5107   × cxp 5657  Rel wrel 5664  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  [cec 8698   / cqs 8699  0cc0 11128  PsMetcpsmet 21575  ~_Metcmetid 34404  pstoMetcpstm 34405

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-po 5567  df-so 5568  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-er 8700  df-ec 8702  df-qs 8706  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-xadd 13168  df-psmet 21583  df-metid 34406  df-pstm 34407

This theorem is used by:  pstmxmet  34415