Theorem pstmfval 33896

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 33895	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}))
3	2	3ad2ant1 1133	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}))
4	3	oveqd 7449	. 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 6919	. . . . 5 ⊢ ∼ ∈ V
6	5	ecelqsi 8814	. . . 4 ⊢ (𝐴 ∈ 𝑋 → [𝐴] ∼ ∈ (𝑋 / ∼ ))
7	6	3ad2ant2 1134	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → [𝐴] ∼ ∈ (𝑋 / ∼ ))
8	5	ecelqsi 8814	. . . 4 ⊢ (𝐵 ∈ 𝑋 → [𝐵] ∼ ∈ (𝑋 / ∼ ))
9	8	3ad2ant3 1135	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → [𝐵] ∼ ∈ (𝑋 / ∼ ))
10		rexeq 3321	. . . . . 6 ⊢ (𝑥 = [𝐴] ∼ → (∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)))
11	10	abbidv 2807	. . . . 5 ⊢ (𝑥 = [𝐴] ∼ → {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)})
12	11	unieqd 4919	. . . 4 ⊢ (𝑥 = [𝐴] ∼ → ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} = ∪ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)})
13		rexeq 3321	. . . . . . 7 ⊢ (𝑦 = [𝐵] ∼ → (∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)))
14	13	rexbidv 3178	. . . . . 6 ⊢ (𝑦 = [𝐵] ∼ → (∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)))
15	14	abbidv 2807	. . . . 5 ⊢ (𝑦 = [𝐵] ∼ → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)})
16	15	unieqd 4919	. . . 4 ⊢ (𝑦 = [𝐵] ∼ → ∪ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} = ∪ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)})
17		eqid 2736	. . . 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 8750	. . . . . . 7 ⊢ ( ∼ ∈ V → [𝐴] ∼ ∈ V)
19	5, 18	ax-mp 5	. . . . . 6 ⊢ [𝐴] ∼ ∈ V
20		ecexg 8750	. . . . . . 7 ⊢ ( ∼ ∈ V → [𝐵] ∼ ∈ V)
21	5, 20	ax-mp 5	. . . . . 6 ⊢ [𝐵] ∼ ∈ V
22	19, 21	ab2rexex 8005	. . . . 5 ⊢ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)} ∈ V
23	22	uniex 7762	. . . 4 ⊢ ∪ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)} ∈ V
24	12, 16, 17, 23	ovmpo 7594	. . 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 584	. 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 1196	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝑒𝐷𝑓))
27		simpl1 1191	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝐷 ∈ (PsMet‘𝑋))
28		simpr1 1194	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝑒 ∈ [𝐴] ∼ )
29		metidss 33891	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) ⊆ (𝑋 × 𝑋))
30	1, 29	eqsstrid 4021	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∼ ⊆ (𝑋 × 𝑋))
31		xpss 5700	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 × 𝑋) ⊆ (V × V)
32	30, 31	sstrdi 3995	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∼ ⊆ (V × V))
33		df-rel 5691	. . . . . . . . . . . . . . . . . 18 ⊢ (Rel ∼ ↔ ∼ ⊆ (V × V))
34	32, 33	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ (PsMet‘𝑋) → Rel ∼ )
35	34	3ad2ant1 1133	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → Rel ∼ )
36	35	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → Rel ∼ )
37		relelec 8793	. . . . . . . . . . . . . . 15 ⊢ (Rel ∼ → (𝑒 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝑒))
38	36, 37	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → (𝑒 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝑒))
39	28, 38	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝐴 ∼ 𝑒)
40	1	breqi 5148	. . . . . . . . . . . . 13 ⊢ (𝐴 ∼ 𝑒 ↔ 𝐴(~Met‘𝐷)𝑒)
41	39, 40	sylib 218	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝐴(~Met‘𝐷)𝑒)
42		simpr2 1195	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝑓 ∈ [𝐵] ∼ )
43		relelec 8793	. . . . . . . . . . . . . . 15 ⊢ (Rel ∼ → (𝑓 ∈ [𝐵] ∼ ↔ 𝐵 ∼ 𝑓))
44	36, 43	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → (𝑓 ∈ [𝐵] ∼ ↔ 𝐵 ∼ 𝑓))
45	42, 44	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝐵 ∼ 𝑓)
46	1	breqi 5148	. . . . . . . . . . . . 13 ⊢ (𝐵 ∼ 𝑓 ↔ 𝐵(~Met‘𝐷)𝑓)
47	45, 46	sylib 218	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝐵(~Met‘𝐷)𝑓)
48		metideq 33893	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝑒 ∧ 𝐵(~Met‘𝐷)𝑓)) → (𝐴𝐷𝐵) = (𝑒𝐷𝑓))
49	27, 41, 47, 48	syl12anc 836	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → (𝐴𝐷𝐵) = (𝑒𝐷𝑓))
50	26, 49	eqtr4d 2779	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑒 ∈ [𝐴] ∼ ∧ 𝑓 ∈ [𝐵] ∼ ∧ 𝑧 = (𝑒𝐷𝑓))) → 𝑧 = (𝐴𝐷𝐵))
51	50	adantlr 715	. . . . . . . . 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 1360	. . . . . . . 8 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)) ∧ 𝑒 ∈ [𝐴] ∼ ) ∧ 𝑓 ∈ [𝐵] ∼ ) ∧ 𝑧 = (𝑒𝐷𝑓)) → 𝑧 = (𝐴𝐷𝐵))
53		oveq1 7439	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑒 → (𝑎𝐷𝑏) = (𝑒𝐷𝑏))
54	53	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑒 → (𝑧 = (𝑎𝐷𝑏) ↔ 𝑧 = (𝑒𝐷𝑏)))
55		oveq2 7440	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑓 → (𝑒𝐷𝑏) = (𝑒𝐷𝑓))
56	55	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑓 → (𝑧 = (𝑒𝐷𝑏) ↔ 𝑧 = (𝑒𝐷𝑓)))
57	54, 56	cbvrex2vw 3241	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏) ↔ ∃𝑒 ∈ [ 𝐴] ∼ ∃𝑓 ∈ [ 𝐵] ∼ 𝑧 = (𝑒𝐷𝑓))
58	57	biimpi 216	. . . . . . . . 9 ⊢ (∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏) → ∃𝑒 ∈ [ 𝐴] ∼ ∃𝑓 ∈ [ 𝐵] ∼ 𝑧 = (𝑒𝐷𝑓))
59	58	adantl 481	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)) → ∃𝑒 ∈ [ 𝐴] ∼ ∃𝑓 ∈ [ 𝐵] ∼ 𝑧 = (𝑒𝐷𝑓))
60	52, 59	r19.29vva 3215	. . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)) → 𝑧 = (𝐴𝐷𝐵))
61		simpl1 1191	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐷 ∈ (PsMet‘𝑋))
62		simpl2 1192	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐴 ∈ 𝑋)
63		psmet0 24319	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0)
64	61, 62, 63	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴𝐷𝐴) = 0)
65		relelec 8793	. . . . . . . . . . 11 ⊢ (Rel ∼ → (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
66	61, 34, 65	3syl 18	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
67	1	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → ∼ = (~Met‘𝐷))
68	67	breqd 5153	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 ∼ 𝐴 ↔ 𝐴(~Met‘𝐷)𝐴))
69		metidv 33892	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐴 ↔ (𝐴𝐷𝐴) = 0))
70	61, 62, 62, 69	syl12anc 836	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴(~Met‘𝐷)𝐴 ↔ (𝐴𝐷𝐴) = 0))
71	66, 68, 70	3bitrd 305	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐴 ∈ [𝐴] ∼ ↔ (𝐴𝐷𝐴) = 0))
72	64, 71	mpbird 257	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐴 ∈ [𝐴] ∼ )
73		simpl3 1193	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐵 ∈ 𝑋)
74		psmet0 24319	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 0)
75	61, 73, 74	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵𝐷𝐵) = 0)
76		relelec 8793	. . . . . . . . . . 11 ⊢ (Rel ∼ → (𝐵 ∈ [𝐵] ∼ ↔ 𝐵 ∼ 𝐵))
77	61, 34, 76	3syl 18	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 ∈ [𝐵] ∼ ↔ 𝐵 ∼ 𝐵))
78	67	breqd 5153	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 ∼ 𝐵 ↔ 𝐵(~Met‘𝐷)𝐵))
79		metidv 33892	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵(~Met‘𝐷)𝐵 ↔ (𝐵𝐷𝐵) = 0))
80	61, 73, 73, 79	syl12anc 836	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵(~Met‘𝐷)𝐵 ↔ (𝐵𝐷𝐵) = 0))
81	77, 78, 80	3bitrd 305	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → (𝐵 ∈ [𝐵] ∼ ↔ (𝐵𝐷𝐵) = 0))
82	75, 81	mpbird 257	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝐵 ∈ [𝐵] ∼ )
83		simpr 484	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → 𝑧 = (𝐴𝐷𝐵))
84		rspceov 7481	. . . . . . . 8 ⊢ ((𝐴 ∈ [𝐴] ∼ ∧ 𝐵 ∈ [𝐵] ∼ ∧ 𝑧 = (𝐴𝐷𝐵)) → ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏))
85	72, 82, 83, 84	syl3anc 1372	. . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑧 = (𝐴𝐷𝐵)) → ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏))
86	60, 85	impbida 800	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏) ↔ 𝑧 = (𝐴𝐷𝐵)))
87	86	abbidv 2807	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)} = {𝑧 ∣ 𝑧 = (𝐴𝐷𝐵)})
88		df-sn 4626	. . . . 5 ⊢ {(𝐴𝐷𝐵)} = {𝑧 ∣ 𝑧 = (𝐴𝐷𝐵)}
89	87, 88	eqtr4di 2794	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)} = {(𝐴𝐷𝐵)})
90	89	unieqd 4919	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∪ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)} = ∪ {(𝐴𝐷𝐵)})
91		ovex 7465	. . . 4 ⊢ (𝐴𝐷𝐵) ∈ V
92	91	unisn 4925	. . 3 ⊢ ∪ {(𝐴𝐷𝐵)} = (𝐴𝐷𝐵)
93	90, 92	eqtrdi 2792	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∪ {𝑧 ∣ ∃𝑎 ∈ [ 𝐴] ∼ ∃𝑏 ∈ [ 𝐵] ∼ 𝑧 = (𝑎𝐷𝑏)} = (𝐴𝐷𝐵))
94	4, 25, 93	3eqtrd 2780	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ([𝐴] ∼ (pstoMet‘𝐷)[𝐵] ∼ ) = (𝐴𝐷𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  {cab 2713  ∃wrex 3069  Vcvv 3479   ⊆ wss 3950  {csn 4625  ∪ cuni 4906   class class class wbr 5142   × cxp 5682  Rel wrel 5689  ‘cfv 6560  (class class class)co 7432   ∈ cmpo 7434  [cec 8744   / cqs 8745  0cc0 11156  PsMetcpsmet 21349  ~_Metcmetid 33886  pstoMetcpstm 33887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-po 5591  df-so 5592  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-er 8746  df-ec 8748  df-qs 8752  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-xadd 13156  df-psmet 21357  df-metid 33888  df-pstm 33889

This theorem is referenced by:  pstmxmet  33897