Theorem restmetu 23836

Description: The uniform structure generated by the restriction of a metric is its trace. (Contributed by Thierry Arnoux, 18-Dec-2017.)

Assertion

Ref	Expression
restmetu	⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = (metUnif‘(𝐷 ↾ (𝐴 × 𝐴))))

Proof of Theorem restmetu

Dummy variables 𝑎 𝑏 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ≠ ∅)
2		psmetres2 23577	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴))
3	2	3adant1 1130	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴))
4		oveq2 7354	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
5	4	imaeq2d 6006	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
6	5	cbvmptv 5213	. . . . . 6 ⊢ (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
7	6	rneqi 5885	. . . . 5 ⊢ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
8	7	metustfbas 23823	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴)) → ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∈ (fBas‘(𝐴 × 𝐴)))
9	1, 3, 8	syl2anc 585	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∈ (fBas‘(𝐴 × 𝐴)))
10		fgval 23131	. . 3 ⊢ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∈ (fBas‘(𝐴 × 𝐴)) → ((𝐴 × 𝐴)filGenran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))) = {𝑣 ∈ 𝒫 (𝐴 × 𝐴) ∣ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) ≠ ∅})
11	9, 10	syl 17	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐴 × 𝐴)filGenran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))) = {𝑣 ∈ 𝒫 (𝐴 × 𝐴) ∣ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) ≠ ∅})
12		metuval 23815	. . 3 ⊢ ((𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴) → (metUnif‘(𝐷 ↾ (𝐴 × 𝐴))) = ((𝐴 × 𝐴)filGenran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))))
13	3, 12	syl 17	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (metUnif‘(𝐷 ↾ (𝐴 × 𝐴))) = ((𝐴 × 𝐴)filGenran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))))
14		fvex 6847	. . . 4 ⊢ (metUnif‘𝐷) ∈ V
15	3	elfvexd 6873	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
16	15, 15	xpexd 7672	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ∈ V)
17		restval 17239	. . . 4 ⊢ (((metUnif‘𝐷) ∈ V ∧ (𝐴 × 𝐴) ∈ V) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))))
18	14, 16, 17	sylancr 588	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))))
19		inss2 4184	. . . . . . . . . . 11 ⊢ (𝑣 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
20		sseq1 3964	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → (𝑢 ⊆ (𝐴 × 𝐴) ↔ (𝑣 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)))
21	19, 20	mpbiri 258	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ⊆ (𝐴 × 𝐴))
22		vex 3447	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
23	22	elpw 4559	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑢 ⊆ (𝐴 × 𝐴))
24	21, 23	sylibr 233	. . . . . . . . 9 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
25	24	rexlimivw 3146	. . . . . . . 8 ⊢ (∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
26	25	adantl 483	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
27		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
28		nfmpt1 5208	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎(𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
29	28	nfrn 5900	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
30	29	nfcri 2892	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
31	27, 30	nfan 1902	. . . . . . . . . . 11 ⊢ Ⅎ𝑎((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
32		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑎 𝑤 ⊆ 𝑣
33	31, 32	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑎(((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣)
34		nfmpt1 5208	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎(𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
35	34	nfrn 5900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
36		nfcv 2905	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎𝒫 𝑢
37	35, 36	nfin 4171	. . . . . . . . . . 11 ⊢ Ⅎ𝑎(ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢)
38		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑎∅
39	37, 38	nfne 3043	. . . . . . . . . 10 ⊢ Ⅎ𝑎(ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅
40		simplr 767	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (◡𝐷 “ (0[,)𝑎))) → 𝑎 ∈ ℝ+)
41		ineq1 4160	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (◡𝐷 “ (0[,)𝑎)) → (𝑤 ∩ (𝐴 × 𝐴)) = ((◡𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
42	41	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (◡𝐷 “ (0[,)𝑎))) → (𝑤 ∩ (𝐴 × 𝐴)) = ((◡𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
43		simp2 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐷 ∈ (PsMet‘𝑋))
44		psmetf 23569	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
45		ffun 6663	. . . . . . . . . . . . . . . 16 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → Fun 𝐷)
46		respreima 7008	. . . . . . . . . . . . . . . 16 ⊢ (Fun 𝐷 → (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = ((◡𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
47	43, 44, 45, 46	4syl 19	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = ((◡𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
48	47	ad6antr 734	. . . . . . . . . . . . . 14 ⊢ ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (◡𝐷 “ (0[,)𝑎))) → (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = ((◡𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
49	42, 48	eqtr4d 2780	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (◡𝐷 “ (0[,)𝑎))) → (𝑤 ∩ (𝐴 × 𝐴)) = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
50		rspe 3230	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℝ+ ∧ (𝑤 ∩ (𝐴 × 𝐴)) = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) → ∃𝑎 ∈ ℝ+ (𝑤 ∩ (𝐴 × 𝐴)) = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
51	40, 49, 50	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (◡𝐷 “ (0[,)𝑎))) → ∃𝑎 ∈ ℝ+ (𝑤 ∩ (𝐴 × 𝐴)) = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
52		vex 3447	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
53	52	inex1 5269	. . . . . . . . . . . . 13 ⊢ (𝑤 ∩ (𝐴 × 𝐴)) ∈ V
54		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
55	54	elrnmpt 5904	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∩ (𝐴 × 𝐴)) ∈ V → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ (𝑤 ∩ (𝐴 × 𝐴)) = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))))
56	53, 55	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑤 ∩ (𝐴 × 𝐴)) ∈ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ (𝑤 ∩ (𝐴 × 𝐴)) = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
57	51, 56	sylibr 233	. . . . . . . . . . 11 ⊢ ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (◡𝐷 “ (0[,)𝑎))) → (𝑤 ∩ (𝐴 × 𝐴)) ∈ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))))
58		simpllr 774	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (◡𝐷 “ (0[,)𝑎))) → 𝑤 ⊆ 𝑣)
59		ssinss1 4192	. . . . . . . . . . . . 13 ⊢ (𝑤 ⊆ 𝑣 → (𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣)
60	58, 59	syl 17	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (◡𝐷 “ (0[,)𝑎))) → (𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣)
61		inss2 4184	. . . . . . . . . . . . 13 ⊢ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
62	61	a1i 11	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (◡𝐷 “ (0[,)𝑎))) → (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴))
63		pweq 4569	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝒫 𝑢 = 𝒫 (𝑣 ∩ (𝐴 × 𝐴)))
64	63	eleq2d 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ (𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 (𝑣 ∩ (𝐴 × 𝐴))))
65	53	elpw 4559	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 (𝑣 ∩ (𝐴 × 𝐴)) ↔ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
66	64, 65	bitrdi 287	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴))))
67		ssin 4185	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣 ∧ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)) ↔ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
68	66, 67	bitr4di 289	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ ((𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣 ∧ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴))))
69	68	ad5antlr 733	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (◡𝐷 “ (0[,)𝑎))) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ ((𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣 ∧ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴))))
70	60, 62, 69	mpbir2and 711	. . . . . . . . . . 11 ⊢ ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (◡𝐷 “ (0[,)𝑎))) → (𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢)
71		inelcm 4419	. . . . . . . . . . 11 ⊢ (((𝑤 ∩ (𝐴 × 𝐴)) ∈ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∧ (𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢) → (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)
72	57, 70, 71	syl2anc 585	. . . . . . . . . 10 ⊢ ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (◡𝐷 “ (0[,)𝑎))) → (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)
73		simplr 767	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) → 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
74		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
75	74	elrnmpt 5904	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑎))))
76	75	elv 3449	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑎)))
77	73, 76	sylib 217	. . . . . . . . . 10 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) → ∃𝑎 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑎)))
78	33, 39, 72, 77	r19.29af2 3248	. . . . . . . . 9 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣) → (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)
79		ssn0 4355	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐴 ≠ ∅) → 𝑋 ≠ ∅)
80	79	ancoms 460	. . . . . . . . . . . . 13 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ 𝑋) → 𝑋 ≠ ∅)
81	80	3adant2 1131	. . . . . . . . . . . 12 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ≠ ∅)
82		metuel 23830	. . . . . . . . . . . 12 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑣 ∈ (metUnif‘𝐷) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑣)))
83	81, 43, 82	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑣 ∈ (metUnif‘𝐷) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑣)))
84	83	simplbda 501	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) → ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑣)
85	84	adantr 482	. . . . . . . . 9 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑣)
86	78, 85	r19.29a 3157	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)
87	86	r19.29an 3153	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)
88	26, 87	jca 513	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅))
89		simprl 769	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
90	89	elpwid 4564	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝑢 ⊆ (𝐴 × 𝐴))
91		simpl3 1193	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝐴 ⊆ 𝑋)
92		xpss12 5642	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
93	91, 91, 92	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
94	90, 93	sstrd 3949	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝑢 ⊆ (𝑋 × 𝑋))
95		difssd 4087	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ⊆ (𝑋 × 𝑋))
96	94, 95	unssd 4141	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ⊆ (𝑋 × 𝑋))
97		simplr 767	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → 𝑏 ∈ ℝ+)
98		eqidd 2738	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → (◡𝐷 “ (0[,)𝑏)) = (◡𝐷 “ (0[,)𝑏)))
99	4	imaeq2d 6006	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑏)))
100	99	rspceeqv 3590	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ℝ+ ∧ (◡𝐷 “ (0[,)𝑏)) = (◡𝐷 “ (0[,)𝑏))) → ∃𝑎 ∈ ℝ+ (◡𝐷 “ (0[,)𝑏)) = (◡𝐷 “ (0[,)𝑎)))
101	97, 98, 100	syl2anc 585	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → ∃𝑎 ∈ ℝ+ (◡𝐷 “ (0[,)𝑏)) = (◡𝐷 “ (0[,)𝑎)))
102	43	ad4antr 730	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → 𝐷 ∈ (PsMet‘𝑋))
103		cnvexg 7848	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
104		imaexg 7839	. . . . . . . . . . . 12 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑏)) ∈ V)
105	74	elrnmpt 5904	. . . . . . . . . . . 12 ⊢ ((◡𝐷 “ (0[,)𝑏)) ∈ V → ((◡𝐷 “ (0[,)𝑏)) ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ (◡𝐷 “ (0[,)𝑏)) = (◡𝐷 “ (0[,)𝑎))))
106	102, 103, 104, 105	4syl 19	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → ((◡𝐷 “ (0[,)𝑏)) ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ (◡𝐷 “ (0[,)𝑏)) = (◡𝐷 “ (0[,)𝑎))))
107	101, 106	mpbird 257	. . . . . . . . . 10 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → (◡𝐷 “ (0[,)𝑏)) ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
108		cnvimass 6026	. . . . . . . . . . . . . . . 16 ⊢ (◡𝐷 “ (0[,)𝑏)) ⊆ dom 𝐷
109	108, 44	fssdm 6680	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (◡𝐷 “ (0[,)𝑏)) ⊆ (𝑋 × 𝑋))
110	102, 109	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → (◡𝐷 “ (0[,)𝑏)) ⊆ (𝑋 × 𝑋))
111		ssdif0 4318	. . . . . . . . . . . . . 14 ⊢ ((◡𝐷 “ (0[,)𝑏)) ⊆ (𝑋 × 𝑋) ↔ ((◡𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) = ∅)
112	110, 111	sylib 217	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → ((◡𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) = ∅)
113		0ss 4351	. . . . . . . . . . . . 13 ⊢ ∅ ⊆ 𝑢
114	112, 113	eqsstrdi 3993	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → ((◡𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ⊆ 𝑢)
115		respreima 7008	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐷 → (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)) = ((◡𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴)))
116	102, 44, 45, 115	4syl 19	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)) = ((◡𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴)))
117		simpr 486	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
118		simpllr 774	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → 𝑣 ∈ 𝒫 𝑢)
119	118	elpwid 4564	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → 𝑣 ⊆ 𝑢)
120	117, 119	eqsstrrd 3978	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)) ⊆ 𝑢)
121	116, 120	eqsstrrd 3978	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → ((◡𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴)) ⊆ 𝑢)
122	114, 121	unssd 4141	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → (((◡𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ∪ ((◡𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
123		ssundif 4440	. . . . . . . . . . . 12 ⊢ ((◡𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ↔ ((◡𝐷 “ (0[,)𝑏)) ∖ 𝑢) ⊆ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)))
124		difcom 4441	. . . . . . . . . . . 12 ⊢ (((◡𝐷 “ (0[,)𝑏)) ∖ 𝑢) ⊆ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ↔ ((◡𝐷 “ (0[,)𝑏)) ∖ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ⊆ 𝑢)
125		difdif2 4241	. . . . . . . . . . . . 13 ⊢ ((◡𝐷 “ (0[,)𝑏)) ∖ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) = (((◡𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ∪ ((◡𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴)))
126	125	sseq1i 3967	. . . . . . . . . . . 12 ⊢ (((◡𝐷 “ (0[,)𝑏)) ∖ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ⊆ 𝑢 ↔ (((◡𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ∪ ((◡𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
127	123, 124, 126	3bitri 297	. . . . . . . . . . 11 ⊢ ((◡𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ↔ (((◡𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ∪ ((◡𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
128	122, 127	sylibr 233	. . . . . . . . . 10 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → (◡𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))))
129		sseq1 3964	. . . . . . . . . . 11 ⊢ (𝑤 = (◡𝐷 “ (0[,)𝑏)) → (𝑤 ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ↔ (◡𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)))))
130	129	rspcev 3576	. . . . . . . . . 10 ⊢ (((◡𝐷 “ (0[,)𝑏)) ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ (◡𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)))) → ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))))
131	107, 128, 130	syl2anc 585	. . . . . . . . 9 ⊢ ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) ∧ 𝑣 ∈ 𝒫 𝑢) ∧ 𝑏 ∈ ℝ+) ∧ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))) → ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))))
132		elin 3921	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ↔ (𝑣 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∧ 𝑣 ∈ 𝒫 𝑢))
133	6	elrnmpt 5904	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ V → (𝑣 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
134	133	elv 3449	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
135	134	anbi1i 625	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∧ 𝑣 ∈ 𝒫 𝑢) ↔ (∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)) ∧ 𝑣 ∈ 𝒫 𝑢))
136		ancom 462	. . . . . . . . . . . . . 14 ⊢ ((∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)) ∧ 𝑣 ∈ 𝒫 𝑢) ↔ (𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
137	132, 135, 136	3bitri 297	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ↔ (𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
138	137	exbii 1850	. . . . . . . . . . . 12 ⊢ (∃𝑣 𝑣 ∈ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ↔ ∃𝑣(𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
139		n0 4301	. . . . . . . . . . . 12 ⊢ ((ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅ ↔ ∃𝑣 𝑣 ∈ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢))
140		df-rex 3072	. . . . . . . . . . . 12 ⊢ (∃𝑣 ∈ 𝒫 𝑢∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)) ↔ ∃𝑣(𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
141	138, 139, 140	3bitr4i 303	. . . . . . . . . . 11 ⊢ ((ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅ ↔ ∃𝑣 ∈ 𝒫 𝑢∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
142	141	biimpi 215	. . . . . . . . . 10 ⊢ ((ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅ → ∃𝑣 ∈ 𝒫 𝑢∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
143	142	ad2antll 727	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → ∃𝑣 ∈ 𝒫 𝑢∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
144	131, 143	r19.29vva 3205	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))))
145	81	adantr 482	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝑋 ≠ ∅)
146	43	adantr 482	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝐷 ∈ (PsMet‘𝑋))
147		metuel 23830	. . . . . . . . 9 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∈ (metUnif‘𝐷) ↔ ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))))))
148	145, 146, 147	syl2anc 585	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∈ (metUnif‘𝐷) ↔ ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))))))
149	96, 144, 148	mpbir2and 711	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∈ (metUnif‘𝐷))
150		indir 4230	. . . . . . . . 9 ⊢ ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)) = ((𝑢 ∩ (𝐴 × 𝐴)) ∪ (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴)))
151		disjdifr 4427	. . . . . . . . . 10 ⊢ (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴)) = ∅
152	151	uneq2i 4115	. . . . . . . . 9 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) ∪ (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴))) = ((𝑢 ∩ (𝐴 × 𝐴)) ∪ ∅)
153		un0 4345	. . . . . . . . 9 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) ∪ ∅) = (𝑢 ∩ (𝐴 × 𝐴))
154	150, 152, 153	3eqtri 2769	. . . . . . . 8 ⊢ ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)) = (𝑢 ∩ (𝐴 × 𝐴))
155		df-ss 3922	. . . . . . . . 9 ⊢ (𝑢 ⊆ (𝐴 × 𝐴) ↔ (𝑢 ∩ (𝐴 × 𝐴)) = 𝑢)
156	90, 155	sylib 217	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → (𝑢 ∩ (𝐴 × 𝐴)) = 𝑢)
157	154, 156	eqtr2id 2790	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝑢 = ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)))
158		ineq1 4160	. . . . . . . 8 ⊢ (𝑣 = (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) → (𝑣 ∩ (𝐴 × 𝐴)) = ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)))
159	158	rspceeqv 3590	. . . . . . 7 ⊢ (((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∈ (metUnif‘𝐷) ∧ 𝑢 = ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴))) → ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
160	149, 157, 159	syl2anc 585	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
161	88, 160	impbida 799	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) ↔ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)))
162		eqid 2737	. . . . . . 7 ⊢ (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) = (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴)))
163	162	elrnmpt 5904	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) ↔ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))))
164	163	elv 3449	. . . . 5 ⊢ (𝑢 ∈ ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) ↔ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
165		pweq 4569	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → 𝒫 𝑣 = 𝒫 𝑢)
166	165	ineq2d 4167	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) = (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢))
167	166	neeq1d 3001	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) ≠ ∅ ↔ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅))
168	167	elrab 3640	. . . . 5 ⊢ (𝑢 ∈ {𝑣 ∈ 𝒫 (𝐴 × 𝐴) ∣ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) ≠ ∅} ↔ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅))
169	161, 164, 168	3bitr4g 314	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑢 ∈ ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) ↔ 𝑢 ∈ {𝑣 ∈ 𝒫 (𝐴 × 𝐴) ∣ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) ≠ ∅}))
170	169	eqrdv 2735	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) = {𝑣 ∈ 𝒫 (𝐴 × 𝐴) ∣ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) ≠ ∅})
171	18, 170	eqtrd 2777	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = {𝑣 ∈ 𝒫 (𝐴 × 𝐴) ∣ (ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) ≠ ∅})
172	11, 13, 171	3eqtr4rd 2788	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = (metUnif‘(𝐷 ↾ (𝐴 × 𝐴))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2941  ∃wrex 3071  {crab 3405  Vcvv 3443   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ∅c0 4277  𝒫 cpw 4555   ↦ cmpt 5183   × cxp 5625  ^◡ccnv 5626  ran crn 5628   ↾ cres 5629   “ cima 5630  Fun wfun 6482  ⟶wf 6484  ‘cfv 6488  (class class class)co 7346  0cc0 10981  ℝ^*cxr 11118  ℝ⁺crp 12840  [,)cico 13191   ↾_t crest 17233  PsMetcpsmet 20691  fBascfbas 20695  filGencfg 20696  metUnifcmetu 20698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5237  ax-sep 5251  ax-nul 5258  ax-pow 5315  ax-pr 5379  ax-un 7659  ax-cnex 11037  ax-resscn 11038  ax-1cn 11039  ax-icn 11040  ax-addcl 11041  ax-addrcl 11042  ax-mulcl 11043  ax-mulrcl 11044  ax-mulcom 11045  ax-addass 11046  ax-mulass 11047  ax-distr 11048  ax-i2m1 11049  ax-1ne0 11050  ax-1rid 11051  ax-rnegex 11052  ax-rrecex 11053  ax-cnre 11054  ax-pre-lttri 11055  ax-pre-lttrn 11056  ax-pre-ltadd 11057  ax-pre-mulgt0 11058

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3735  df-csb 3851  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4278  df-if 4482  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4861  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5184  df-id 5525  df-po 5539  df-so 5540  df-xp 5633  df-rel 5634  df-cnv 5635  df-co 5636  df-dm 5637  df-rn 5638  df-res 5639  df-ima 5640  df-iota 6440  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7302  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7908  df-2nd 7909  df-er 8578  df-map 8697  df-en 8814  df-dom 8815  df-sdom 8816  df-pnf 11121  df-mnf 11122  df-xr 11123  df-ltxr 11124  df-le 11125  df-sub 11317  df-neg 11318  df-rp 12841  df-ico 13195  df-rest 17235  df-psmet 20699  df-fbas 20704  df-fg 20705  df-metu 20706

This theorem is referenced by:  reust  24655  qqhucn  32304