Theorem restmetu 24688

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 1152	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ≠ ∅)
2		psmetres2 24432	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴))
3	2	3adant1 1146	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴))
4		oveq2 7408	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
5	4	imaeq2d 6053	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
6	5	cbvmptv 5209	. . . . . 6 ⊢ (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
7	6	rneqi 5918	. . . . 5 ⊢ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
8	7	metustfbas 24675	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴)) → ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∈ (fBas‘(𝐴 × 𝐴)))
9	1, 3, 8	syl2anc 595	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∈ (fBas‘(𝐴 × 𝐴)))
10		fgval 23988	. . 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 18	. 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 24667	. . 3 ⊢ ((𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴) → (metUnif‘(𝐷 ↾ (𝐴 × 𝐴))) = ((𝐴 × 𝐴)filGenran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))))
13	3, 12	syl 18	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (metUnif‘(𝐷 ↾ (𝐴 × 𝐴))) = ((𝐴 × 𝐴)filGenran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))))
14		fvex 6884	. . . 4 ⊢ (metUnif‘𝐷) ∈ V
15	3	elfvexd 6907	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
16	15, 15	xpexd 7738	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ∈ V)
17		restval 17469	. . . 4 ⊢ (((metUnif‘𝐷) ∈ V ∧ (𝐴 × 𝐴) ∈ V) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))))
18	14, 16, 17	sylancr 598	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))))
19		inss2 4192	. . . . . . . . . . 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 261	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ⊆ (𝐴 × 𝐴))
22		vex 3461	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
23	22	elpw 4562	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑢 ⊆ (𝐴 × 𝐴))
24	21, 23	sylibr 237	. . . . . . . . 9 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
25	24	rexlimivw 3162	. . . . . . . 8 ⊢ (∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
26	25	adantl 486	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
27		nfv 1937	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
28		nfmpt1 5204	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎(𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
29	28	nfrn 5933	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
30	29	nfcri 2919	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
31	27, 30	nfan 1922	. . . . . . . . . . 11 ⊢ Ⅎ𝑎((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
32		nfv 1937	. . . . . . . . . . 11 ⊢ Ⅎ𝑎 𝑤 ⊆ 𝑣
33	31, 32	nfan 1922	. . . . . . . . . 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 5204	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎(𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
35	34	nfrn 5933	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
36		nfcv 2927	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎𝒫 𝑢
37	35, 36	nfin 4179	. . . . . . . . . . 11 ⊢ Ⅎ𝑎(ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢)
38		nfcv 2927	. . . . . . . . . . 11 ⊢ Ⅎ𝑎∅
39	37, 38	nfne 3061	. . . . . . . . . 10 ⊢ Ⅎ𝑎(ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅
40		simplr 780	. . . . . . . . . . . . 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 4168	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (◡𝐷 “ (0[,)𝑎)) → (𝑤 ∩ (𝐴 × 𝐴)) = ((◡𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
42	41	adantl 486	. . . . . . . . . . . . . 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 1153	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐷 ∈ (PsMet‘𝑋))
44		psmetf 24424	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
45		ffun 6698	. . . . . . . . . . . . . . . 16 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → Fun 𝐷)
46		respreima 7051	. . . . . . . . . . . . . . . 16 ⊢ (Fun 𝐷 → (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = ((◡𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
47	43, 44, 45, 46	4syl 20	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = ((◡𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
48	47	ad6antr 748	. . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . 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 3255	. . . . . . . . . . . . 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 595	. . . . . . . . . . . 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 3461	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
53	52	inex1 5278	. . . . . . . . . . . . 13 ⊢ (𝑤 ∩ (𝐴 × 𝐴)) ∈ V
54		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
55	54	elrnmpt 5939	. . . . . . . . . . . . 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 237	. . . . . . . . . . 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 787	. . . . . . . . . . . . 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 4200	. . . . . . . . . . . . 13 ⊢ (𝑤 ⊆ 𝑣 → (𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣)
60	58, 59	syl 18	. . . . . . . . . . . 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 4192	. . . . . . . . . . . . 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 4572	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝒫 𝑢 = 𝒫 (𝑣 ∩ (𝐴 × 𝐴)))
64	63	eleq2d 2851	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ (𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 (𝑣 ∩ (𝐴 × 𝐴))))
65	53	elpw 4562	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 (𝑣 ∩ (𝐴 × 𝐴)) ↔ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
66	64, 65	bitrdi 290	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴))))
67		ssin 4193	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣 ∧ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)) ↔ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
68	66, 67	bitr4di 292	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ ((𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣 ∧ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴))))
69	68	ad5antlr 747	. . . . . . . . . . . 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 725	. . . . . . . . . . 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 4422	. . . . . . . . . . 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 595	. . . . . . . . . 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 780	. . . . . . . . . . 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 2765	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
75	74	elrnmpt 5939	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑎))))
76	75	elv 3462	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑎)))
77	73, 76	sylib 221	. . . . . . . . . 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 3273	. . . . . . . . 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 4361	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐴 ≠ ∅) → 𝑋 ≠ ∅)
80	79	ancoms 463	. . . . . . . . . . . . 13 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ 𝑋) → 𝑋 ≠ ∅)
81	80	3adant2 1147	. . . . . . . . . . . 12 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ≠ ∅)
82		metuel 24682	. . . . . . . . . . . 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 595	. . . . . . . . . . 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 504	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) → ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑣)
85	84	adantr 485	. . . . . . . . 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 3173	. . . . . . . 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 3169	. . . . . . 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 520	. . . . . 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 782	. . . . . . . . . . 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 4567	. . . . . . . . . 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 1210	. . . . . . . . . . 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 5667	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
93	91, 91, 92	syl2anc 595	. . . . . . . . . 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 4093	. . . . . . . . 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 4147	. . . . . . . 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 780	. . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 6053	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑏)))
100	99	rspceeqv 3607	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ℝ+ ∧ (◡𝐷 “ (0[,)𝑏)) = (◡𝐷 “ (0[,)𝑏))) → ∃𝑎 ∈ ℝ+ (◡𝐷 “ (0[,)𝑏)) = (◡𝐷 “ (0[,)𝑎)))
101	97, 98, 100	syl2anc 595	. . . . . . . . . . 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 744	. . . . . . . . . . . 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 7909	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
104		imaexg 7898	. . . . . . . . . . . 12 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑏)) ∈ V)
105	74	elrnmpt 5939	. . . . . . . . . . . 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 20	. . . . . . . . . . 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 260	. . . . . . . . . 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 6075	. . . . . . . . . . . . . . . 16 ⊢ (◡𝐷 “ (0[,)𝑏)) ⊆ dom 𝐷
109	108, 44	fssdm 6715	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (◡𝐷 “ (0[,)𝑏)) ⊆ (𝑋 × 𝑋))
110	102, 109	syl 18	. . . . . . . . . . . . . 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 4322	. . . . . . . . . . . . . 14 ⊢ ((◡𝐷 “ (0[,)𝑏)) ⊆ (𝑋 × 𝑋) ↔ ((◡𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) = ∅)
112	110, 111	sylib 221	. . . . . . . . . . . . 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 4357	. . . . . . . . . . . . 13 ⊢ ∅ ⊆ 𝑢
114	112, 113	eqsstrdi 3983	. . . . . . . . . . . 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 7051	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐷 → (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)) = ((◡𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴)))
116	102, 44, 45, 115	4syl 20	. . . . . . . . . . . . 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 489	. . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . 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 4567	. . . . . . . . . . . . . 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 3974	. . . . . . . . . . . . 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 3974	. . . . . . . . . . . 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 4147	. . . . . . . . . . 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 4444	. . . . . . . . . . . 12 ⊢ ((◡𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ↔ ((◡𝐷 “ (0[,)𝑏)) ∖ 𝑢) ⊆ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)))
124		difcom 4445	. . . . . . . . . . . 12 ⊢ (((◡𝐷 “ (0[,)𝑏)) ∖ 𝑢) ⊆ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ↔ ((◡𝐷 “ (0[,)𝑏)) ∖ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ⊆ 𝑢)
125		difdif2 4251	. . . . . . . . . . . . 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 300	. . . . . . . . . . 11 ⊢ ((◡𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ↔ (((◡𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ∪ ((◡𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
128	122, 127	sylibr 237	. . . . . . . . . 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 3584	. . . . . . . . . 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 595	. . . . . . . . 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 3923	. . . . . . . . . . . . . 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 5939	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ V → (𝑣 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
134	133	elv 3462	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
135	134	anbi1i 635	. . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . 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 300	. . . . . . . . . . . . 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 1871	. . . . . . . . . . . 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 4308	. . . . . . . . . . . 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 3090	. . . . . . . . . . . 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 306	. . . . . . . . . . 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 219	. . . . . . . . . 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 741	. . . . . . . . 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 3225	. . . . . . . 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 485	. . . . . . . . 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 485	. . . . . . . . 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 24682	. . . . . . . . 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 595	. . . . . . . 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 725	. . . . . . 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 4241	. . . . . . . . 9 ⊢ ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)) = ((𝑢 ∩ (𝐴 × 𝐴)) ∪ (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴)))
151		disjdifr 4430	. . . . . . . . . 10 ⊢ (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴)) = ∅
152	151	uneq2i 4121	. . . . . . . . 9 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) ∪ (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴))) = ((𝑢 ∩ (𝐴 × 𝐴)) ∪ ∅)
153		un0 4351	. . . . . . . . 9 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) ∪ ∅) = (𝑢 ∩ (𝐴 × 𝐴))
154	150, 152, 153	3eqtri 2792	. . . . . . . 8 ⊢ ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)) = (𝑢 ∩ (𝐴 × 𝐴))
155		dfss2 3925	. . . . . . . . 9 ⊢ (𝑢 ⊆ (𝐴 × 𝐴) ↔ (𝑢 ∩ (𝐴 × 𝐴)) = 𝑢)
156	90, 155	sylib 221	. . . . . . . 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 2813	. . . . . . 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 4168	. . . . . . . 8 ⊢ (𝑣 = (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) → (𝑣 ∩ (𝐴 × 𝐴)) = ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)))
159	158	rspceeqv 3607	. . . . . . 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 595	. . . . . 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 812	. . . . 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 2765	. . . . . . 7 ⊢ (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) = (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴)))
163	162	elrnmpt 5939	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) ↔ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))))
164	163	elv 3462	. . . . 5 ⊢ (𝑢 ∈ ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) ↔ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
165		pweq 4572	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → 𝒫 𝑣 = 𝒫 𝑢)
166	165	ineq2d 4175	. . . . . . 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 3019	. . . . . 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 3653	. . . . 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 317	. . . 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 2763	. . 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 2800	. 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 2811	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = (metUnif‘(𝐷 ↾ (𝐴 × 𝐴))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145   ≠ wne 2960  ∃wrex 3089  {crab 3417  Vcvv 3457   ∖ cdif 3904   ∪ cun 3905   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558   ↦ cmpt 5186   × cxp 5650  ^◡ccnv 5651  ran crn 5653   ↾ cres 5654   “ cima 5655  Fun wfun 6519  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  0cc0 11088  ℝ^*cxr 11230  ℝ⁺crp 13007  [,)cico 13365   ↾_t crest 17463  PsMetcpsmet 21466  fBascfbas 21470  filGencfg 21471  metUnifcmetu 21473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-po 5560  df-so 5561  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-er 8682  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-rp 13008  df-ico 13369  df-rest 17465  df-psmet 21474  df-fbas 21479  df-fg 21480  df-metu 21481

This theorem is referenced by:  reust  25501  qqhucn  34299