Theorem restmetu 24869

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 1154	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ≠ ∅)
2		psmetres2 24613	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴))
3	2	3adant1 1148	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴))
4		oveq2 7420	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
5	4	imaeq2d 6054	. . . . . . 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 5919	. . . . 5 ⊢ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
8	7	metustfbas 24856	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴)) → ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∈ (fBas‘(𝐴 × 𝐴)))
9	1, 3, 8	syl2anc 596	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∈ (fBas‘(𝐴 × 𝐴)))
10		fgval 24169	. . 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 24848	. . 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 6890	. . . 4 ⊢ (metUnif‘𝐷) ∈ V
15	3	elfvexd 6913	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
16	15, 15	xpexd 7754	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ∈ V)
17		restval 17577	. . . 4 ⊢ (((metUnif‘𝐷) ∈ V ∧ (𝐴 × 𝐴) ∈ V) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))))
18	14, 16, 17	sylancr 599	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))))
19		inss2 4183	. . . . . . . . . . 11 ⊢ (𝑣 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
20		sseq1 3956	. . . . . . . . . . 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 3455	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
23	22	elpw 4561	. . . . . . . . . 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 3160	. . . . . . . 8 ⊢ (∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
26	25	adantl 487	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
27		nfv 1947	. . . . . . . . . . . 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 5934	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
30	29	nfcri 2915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
31	27, 30	nfan 1932	. . . . . . . . . . 11 ⊢ Ⅎ𝑎((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
32		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑎 𝑤 ⊆ 𝑣
33	31, 32	nfan 1932	. . . . . . . . . 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 5934	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
36		nfcv 2923	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎𝒫 𝑢
37	35, 36	nfin 4170	. . . . . . . . . . 11 ⊢ Ⅎ𝑎(ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢)
38		nfcv 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑎∅
39	37, 38	nfne 3059	. . . . . . . . . 10 ⊢ Ⅎ𝑎(ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅
40		simplr 781	. . . . . . . . . . . . 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 4159	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (◡𝐷 “ (0[,)𝑎)) → (𝑤 ∩ (𝐴 × 𝐴)) = ((◡𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
42	41	adantl 487	. . . . . . . . . . . . . 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 1155	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐷 ∈ (PsMet‘𝑋))
44		psmetf 24605	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
45		ffun 6704	. . . . . . . . . . . . . . . 16 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → Fun 𝐷)
46		respreima 7057	. . . . . . . . . . . . . . . 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 749	. . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . 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 3253	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 3455	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
53	52	inex1 5277	. . . . . . . . . . . . 13 ⊢ (𝑤 ∩ (𝐴 × 𝐴)) ∈ V
54		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
55	54	elrnmpt 5940	. . . . . . . . . . . . 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 788	. . . . . . . . . . . . 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 4191	. . . . . . . . . . . . 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 4183	. . . . . . . . . . . . 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 4571	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝒫 𝑢 = 𝒫 (𝑣 ∩ (𝐴 × 𝐴)))
64	63	eleq2d 2847	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ (𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 (𝑣 ∩ (𝐴 × 𝐴))))
65	53	elpw 4561	. . . . . . . . . . . . . . 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 4184	. . . . . . . . . . . . . 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 748	. . . . . . . . . . . 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 726	. . . . . . . . . . 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 4418	. . . . . . . . . . 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 596	. . . . . . . . . 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 781	. . . . . . . . . . 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 2761	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
75	74	elrnmpt 5940	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑎))))
76	75	elv 3456	. . . . . . . . . . 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 3271	. . . . . . . . 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 464	. . . . . . . . . . . . 13 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ 𝑋) → 𝑋 ≠ ∅)
81	80	3adant2 1149	. . . . . . . . . . . 12 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ≠ ∅)
82		metuel 24863	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 505	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) → ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑣)
85	84	adantr 486	. . . . . . . . 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 3171	. . . . . . . 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 3167	. . . . . . 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 521	. . . . . 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 783	. . . . . . . . . . 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 4566	. . . . . . . . . 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 1212	. . . . . . . . . . 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 5666	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
93	91, 91, 92	syl2anc 596	. . . . . . . . . 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 3941	. . . . . . . . 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 4084	. . . . . . . . 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 4138	. . . . . . . 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 781	. . . . . . . . . . . 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 2762	. . . . . . . . . . . 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 6054	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑏)))
100	99	rspceeqv 3599	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ℝ+ ∧ (◡𝐷 “ (0[,)𝑏)) = (◡𝐷 “ (0[,)𝑏))) → ∃𝑎 ∈ ℝ+ (◡𝐷 “ (0[,)𝑏)) = (◡𝐷 “ (0[,)𝑎)))
101	97, 98, 100	syl2anc 596	. . . . . . . . . . 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 745	. . . . . . . . . . . 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 7925	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
104		imaexg 7914	. . . . . . . . . . . 12 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑏)) ∈ V)
105	74	elrnmpt 5940	. . . . . . . . . . . 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 6076	. . . . . . . . . . . . . . . 16 ⊢ (◡𝐷 “ (0[,)𝑏)) ⊆ dom 𝐷
109	108, 44	fssdm 6721	. . . . . . . . . . . . . . 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 4314	. . . . . . . . . . . . . 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 4350	. . . . . . . . . . . . 13 ⊢ ∅ ⊆ 𝑢
114	112, 113	eqsstrdi 3975	. . . . . . . . . . . 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 7057	. . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . 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 4566	. . . . . . . . . . . . . 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 3966	. . . . . . . . . . . . 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 3966	. . . . . . . . . . . 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 4138	. . . . . . . . . . 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 4443	. . . . . . . . . . . 12 ⊢ ((◡𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ↔ ((◡𝐷 “ (0[,)𝑏)) ∖ 𝑢) ⊆ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)))
124		difcom 4444	. . . . . . . . . . . 12 ⊢ (((◡𝐷 “ (0[,)𝑏)) ∖ 𝑢) ⊆ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ↔ ((◡𝐷 “ (0[,)𝑏)) ∖ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ⊆ 𝑢)
125		difdif2 4242	. . . . . . . . . . . . 13 ⊢ ((◡𝐷 “ (0[,)𝑏)) ∖ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) = (((◡𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ∪ ((◡𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴)))
126	125	sseq1i 3959	. . . . . . . . . . . 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 3956	. . . . . . . . . . 11 ⊢ (𝑤 = (◡𝐷 “ (0[,)𝑏)) → (𝑤 ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ↔ (◡𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)))))
130	129	rspcev 3577	. . . . . . . . . 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 596	. . . . . . . . 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 3915	. . . . . . . . . . . . . 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 5940	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ V → (𝑣 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
134	133	elv 3456	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑏 ∈ ℝ+ 𝑣 = (◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
135	134	anbi1i 636	. . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . 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 1881	. . . . . . . . . . . 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 4300	. . . . . . . . . . . 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 3088	. . . . . . . . . . . 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 742	. . . . . . . . 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 3223	. . . . . . . 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 486	. . . . . . . . 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 486	. . . . . . . . 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 24863	. . . . . . . . 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 596	. . . . . . . 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 726	. . . . . . 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 4232	. . . . . . . . 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 4112	. . . . . . . . 9 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) ∪ (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴))) = ((𝑢 ∩ (𝐴 × 𝐴)) ∪ ∅)
153		un0 4344	. . . . . . . . 9 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) ∪ ∅) = (𝑢 ∩ (𝐴 × 𝐴))
154	150, 152, 153	3eqtri 2788	. . . . . . . 8 ⊢ ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)) = (𝑢 ∩ (𝐴 × 𝐴))
155		dfss2 3917	. . . . . . . . 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 2809	. . . . . . 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 4159	. . . . . . . 8 ⊢ (𝑣 = (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) → (𝑣 ∩ (𝐴 × 𝐴)) = ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)))
159	158	rspceeqv 3599	. . . . . . 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 596	. . . . . 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 813	. . . . 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 2761	. . . . . . 7 ⊢ (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) = (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴)))
163	162	elrnmpt 5940	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) ↔ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))))
164	163	elv 3456	. . . . 5 ⊢ (𝑢 ∈ ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) ↔ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
165		pweq 4571	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → 𝒫 𝑣 = 𝒫 𝑢)
166	165	ineq2d 4166	. . . . . . 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 3015	. . . . . 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 3645	. . . . 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 2759	. . 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 2796	. 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 2807	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = (metUnif‘(𝐷 ↾ (𝐴 × 𝐴))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087  {crab 3413  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557   ↦ cmpt 5186   × cxp 5649  ^◡ccnv 5650  ran crn 5652   ↾ cres 5653   “ cima 5654  Fun wfun 6525  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  0cc0 11181  ℝ^*cxr 11323  ℝ⁺crp 13101  [,)cico 13459   ↾_t crest 17571  PsMetcpsmet 21642  fBascfbas 21646  filGencfg 21647  metUnifcmetu 21649

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-so 5560  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-rp 13102  df-ico 13463  df-rest 17573  df-psmet 21650  df-fbas 21655  df-fg 21656  df-metu 21657

This theorem is used by:  reust  25682  qqhucn  34606