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Theorem restmetu 22185

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 1054	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ≠ ∅)
2		psmetres2 21929	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴))
3	2	3adant1 1072	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴))
4		oveq2 6557	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
5	4	imaeq2d 5385	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
6	5	cbvmptv 4678	. . . . . 6 ⊢ (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = (𝑏 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
7	6	rneqi 5273	. . . . 5 ⊢ ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
8	7	metustfbas 22172	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴)) → ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∈ (fBas‘(𝐴 × 𝐴)))
9	1, 3, 8	syl2anc 691	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∈ (fBas‘(𝐴 × 𝐴)))
10		fgval 21484	. . 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 22164	. . 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 6113	. . . 4 ⊢ (metUnif‘𝐷) ∈ V
15	3	elfvexd 6132	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
16		xpexg 6858	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 × 𝐴) ∈ V)
17	15, 15, 16	syl2anc 691	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ∈ V)
18		restval 15910	. . . 4 ⊢ (((metUnif‘𝐷) ∈ V ∧ (𝐴 × 𝐴) ∈ V) → ((metUnif‘𝐷) ↾_t (𝐴 × 𝐴)) = ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))))
19	14, 17, 18	sylancr 694	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((metUnif‘𝐷) ↾_t (𝐴 × 𝐴)) = ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))))
20		inss2 3796	. . . . . . . . . . 11 ⊢ (𝑣 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
21		sseq1 3589	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → (𝑢 ⊆ (𝐴 × 𝐴) ↔ (𝑣 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)))
22	20, 21	mpbiri 247	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ⊆ (𝐴 × 𝐴))
23		vex 3176	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
24	23	elpw 4114	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑢 ⊆ (𝐴 × 𝐴))
25	22, 24	sylibr 223	. . . . . . . . 9 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
26	25	rexlimivw 3011	. . . . . . . 8 ⊢ (∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
27	26	adantl 481	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
28		nfv 1830	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
29		nfmpt1 4675	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎(𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))
30	29	nfrn 5289	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))
31	30	nfcri 2745	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎 𝑤 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))
32	28, 31	nfan 1816	. . . . . . . . . . 11 ⊢ Ⅎ𝑎((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎))))
33		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑎 𝑤 ⊆ 𝑣
34	32, 33	nfan 1816	. . . . . . . . . 10 ⊢ Ⅎ𝑎(((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))) ∧ 𝑤 ⊆ 𝑣)
35		nfmpt1 4675	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎(𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
36	35	nfrn 5289	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
37		nfcv 2751	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎𝒫 𝑢
38	36, 37	nfin 3782	. . . . . . . . . . 11 ⊢ Ⅎ𝑎(ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢)
39		nfcv 2751	. . . . . . . . . . 11 ⊢ Ⅎ𝑎∅
40	38, 39	nfne 2882	. . . . . . . . . 10 ⊢ Ⅎ𝑎(ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅
41		simplr 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))) → 𝑎 ∈ ℝ⁺)
42		ineq1 3769	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (^◡𝐷 “ (0[,)𝑎)) → (𝑤 ∩ (𝐴 × 𝐴)) = ((^◡𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
43	42	adantl 481	. . . . . . . . . . . . . 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)))
44		simp2 1055	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐷 ∈ (PsMet‘𝑋))
45		psmetf 21921	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
46		ffun 5961	. . . . . . . . . . . . . . . 16 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ^* → Fun 𝐷)
47		respreima 6252	. . . . . . . . . . . . . . . 16 ⊢ (Fun 𝐷 → (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = ((^◡𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
48	44, 45, 46, 47	4syl 19	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = ((^◡𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
49	48	ad6antr 768	. . . . . . . . . . . . . 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)))
50	43, 49	eqtr4d 2647	. . . . . . . . . . . . 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)))
51		rspe 2986	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℝ⁺ ∧ (𝑤 ∩ (𝐴 × 𝐴)) = (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) → ∃𝑎 ∈ ℝ⁺ (𝑤 ∩ (𝐴 × 𝐴)) = (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
52	41, 50, 51	syl2anc 691	. . . . . . . . . . . 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)))
53		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
54	53	inex1 4727	. . . . . . . . . . . . 13 ⊢ (𝑤 ∩ (𝐴 × 𝐴)) ∈ V
55		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
56	55	elrnmpt 5293	. . . . . . . . . . . . 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))))
57	54, 56	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)))
58	52, 57	sylibr 223	. . . . . . . . . . 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))))
59		simpllr 795	. . . . . . . . . . . . 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)
60		ssinss1 3803	. . . . . . . . . . . . 13 ⊢ (𝑤 ⊆ 𝑣 → (𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣)
61	59, 60	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)
62		inss2 3796	. . . . . . . . . . . . 13 ⊢ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
63	62	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))
64		pweq 4111	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝒫 𝑢 = 𝒫 (𝑣 ∩ (𝐴 × 𝐴)))
65	64	eleq2d 2673	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ (𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 (𝑣 ∩ (𝐴 × 𝐴))))
66	54	elpw 4114	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 (𝑣 ∩ (𝐴 × 𝐴)) ↔ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
67	65, 66	syl6bb 275	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴))))
68		ssin 3797	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣 ∧ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)) ↔ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
69	67, 68	syl6bbr 277	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ ((𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣 ∧ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴))))
70	69	ad5antlr 767	. . . . . . . . . . . 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))))
71	61, 63, 70	mpbir2and 959	. . . . . . . . . . 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)
72		inelcm 3984	. . . . . . . . . . 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) ≠ ∅)
73	58, 71, 72	syl2anc 691	. . . . . . . . . 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) ≠ ∅)
74		simplr 788	. . . . . . . . . . 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))))
75		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))
76	75	elrnmpt 5293	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ⁺ 𝑤 = (^◡𝐷 “ (0[,)𝑎))))
77	53, 76	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ⁺ 𝑤 = (^◡𝐷 “ (0[,)𝑎)))
78	74, 77	sylib 207	. . . . . . . . . 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)))
79	34, 40, 73, 78	r19.29af2 3057	. . . . . . . . 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) ≠ ∅)
80		ssn0 3928	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐴 ≠ ∅) → 𝑋 ≠ ∅)
81	80	ancoms 468	. . . . . . . . . . . . 13 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ 𝑋) → 𝑋 ≠ ∅)
82	81	3adant2 1073	. . . . . . . . . . . 12 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ≠ ∅)
83		metuel 22179	. . . . . . . . . . . 12 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑣 ∈ (metUnif‘𝐷) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑣)))
84	82, 44, 83	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑣 ∈ (metUnif‘𝐷) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑣)))
85	84	simplbda 652	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) → ∃𝑤 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑣)
86	85	adantr 480	. . . . . . . . 9 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → ∃𝑤 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑣)
87	79, 86	r19.29a 3060	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → (ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)
88	87	r19.29an 3059	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → (ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)
89	27, 88	jca 553	. . . . . 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) ≠ ∅))
90		simprl 790	. . . . . . . . . . 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))
91	90	elpwid 4118	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝑢 ⊆ (𝐴 × 𝐴))
92		simpl3 1059	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝐴 ⊆ 𝑋)
93		xpss12 5148	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝑋) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
94	92, 92, 93	syl2anc 691	. . . . . . . . . 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))
95	91, 94	sstrd 3578	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝑢 ⊆ (𝑋 × 𝑋))
96		difssd 3700	. . . . . . . . 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))
97	95, 96	unssd 3751	. . . . . . . 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))
98		simplr 788	. . . . . . . . . . . 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⁺)
99		eqidd 2611	. . . . . . . . . . . 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)))
100	4	imaeq2d 5385	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (^◡𝐷 “ (0[,)𝑎)) = (^◡𝐷 “ (0[,)𝑏)))
101	100	eqeq2d 2620	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → ((^◡𝐷 “ (0[,)𝑏)) = (^◡𝐷 “ (0[,)𝑎)) ↔ (^◡𝐷 “ (0[,)𝑏)) = (^◡𝐷 “ (0[,)𝑏))))
102	101	rspcev 3282	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ℝ⁺ ∧ (^◡𝐷 “ (0[,)𝑏)) = (^◡𝐷 “ (0[,)𝑏))) → ∃𝑎 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑏)) = (^◡𝐷 “ (0[,)𝑎)))
103	98, 99, 102	syl2anc 691	. . . . . . . . . . 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)))
104	44	ad4antr 764	. . . . . . . . . . . 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))
105		cnvexg 7005	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ^◡𝐷 ∈ V)
106		imaexg 6995	. . . . . . . . . . . 12 ⊢ (^◡𝐷 ∈ V → (^◡𝐷 “ (0[,)𝑏)) ∈ V)
107	75	elrnmpt 5293	. . . . . . . . . . . 12 ⊢ ((^◡𝐷 “ (0[,)𝑏)) ∈ V → ((^◡𝐷 “ (0[,)𝑏)) ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ⁺ (^◡𝐷 “ (0[,)𝑏)) = (^◡𝐷 “ (0[,)𝑎))))
108	104, 105, 106, 107	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))))
109	103, 108	mpbird 246	. . . . . . . . . 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))))
110		cnvimass 5404	. . . . . . . . . . . . . . . 16 ⊢ (^◡𝐷 “ (0[,)𝑏)) ⊆ dom 𝐷
111		fdm 5964	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ^* → dom 𝐷 = (𝑋 × 𝑋))
112	45, 111	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
113	110, 112	syl5sseq 3616	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (^◡𝐷 “ (0[,)𝑏)) ⊆ (𝑋 × 𝑋))
114	104, 113	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))
115		ssdif0 3896	. . . . . . . . . . . . . 14 ⊢ ((^◡𝐷 “ (0[,)𝑏)) ⊆ (𝑋 × 𝑋) ↔ ((^◡𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) = ∅)
116	114, 115	sylib 207	. . . . . . . . . . . . 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)) = ∅)
117		0ss 3924	. . . . . . . . . . . . 13 ⊢ ∅ ⊆ 𝑢
118	116, 117	syl6eqss 3618	. . . . . . . . . . . 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)
119		respreima 6252	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐷 → (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)) = ((^◡𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴)))
120	104, 45, 46, 119	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)))
121		simpr 476	. . . . . . . . . . . . . 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)))
122		simpllr 795	. . . . . . . . . . . . . . 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)
123	122	elpwid 4118	. . . . . . . . . . . . . 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)
124	121, 123	eqsstr3d 3603	. . . . . . . . . . . . 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)
125	120, 124	eqsstr3d 3603	. . . . . . . . . . . 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)
126	118, 125	unssd 3751	. . . . . . . . . . 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)
127		ssundif 4004	. . . . . . . . . . . 12 ⊢ ((^◡𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ↔ ((^◡𝐷 “ (0[,)𝑏)) ∖ 𝑢) ⊆ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)))
128		difcom 4005	. . . . . . . . . . . 12 ⊢ (((^◡𝐷 “ (0[,)𝑏)) ∖ 𝑢) ⊆ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ↔ ((^◡𝐷 “ (0[,)𝑏)) ∖ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ⊆ 𝑢)
129		difdif2 3843	. . . . . . . . . . . . 13 ⊢ ((^◡𝐷 “ (0[,)𝑏)) ∖ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) = (((^◡𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ∪ ((^◡𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴)))
130	129	sseq1i 3592	. . . . . . . . . . . 12 ⊢ (((^◡𝐷 “ (0[,)𝑏)) ∖ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ⊆ 𝑢 ↔ (((^◡𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ∪ ((^◡𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
131	127, 128, 130	3bitri 285	. . . . . . . . . . 11 ⊢ ((^◡𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ↔ (((^◡𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ∪ ((^◡𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
132	126, 131	sylibr 223	. . . . . . . . . 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))))
133		sseq1 3589	. . . . . . . . . . 11 ⊢ (𝑤 = (^◡𝐷 “ (0[,)𝑏)) → (𝑤 ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ↔ (^◡𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)))))
134	133	rspcev 3282	. . . . . . . . . 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))))
135	109, 132, 134	syl2anc 691	. . . . . . . . 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))))
136		elin 3758	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ↔ (𝑣 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∧ 𝑣 ∈ 𝒫 𝑢))
137		vex 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑣 ∈ V
138	6	elrnmpt 5293	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ V → (𝑣 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑏 ∈ ℝ⁺ 𝑣 = (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
139	137, 138	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑏 ∈ ℝ⁺ 𝑣 = (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
140	139	anbi1i 727	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∧ 𝑣 ∈ 𝒫 𝑢) ↔ (∃𝑏 ∈ ℝ⁺ 𝑣 = (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)) ∧ 𝑣 ∈ 𝒫 𝑢))
141		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))))
142	136, 140, 141	3bitri 285	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ↔ (𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑏 ∈ ℝ⁺ 𝑣 = (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
143	142	exbii 1764	. . . . . . . . . . . 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))))
144		n0 3890	. . . . . . . . . . . 12 ⊢ ((ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅ ↔ ∃𝑣 𝑣 ∈ (ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢))
145		df-rex 2902	. . . . . . . . . . . 12 ⊢ (∃𝑣 ∈ 𝒫 𝑢∃𝑏 ∈ ℝ⁺ 𝑣 = (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)) ↔ ∃𝑣(𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑏 ∈ ℝ⁺ 𝑣 = (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
146	143, 144, 145	3bitr4i 291	. . . . . . . . . . 11 ⊢ ((ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅ ↔ ∃𝑣 ∈ 𝒫 𝑢∃𝑏 ∈ ℝ⁺ 𝑣 = (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
147	146	biimpi 205	. . . . . . . . . 10 ⊢ ((ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅ → ∃𝑣 ∈ 𝒫 𝑢∃𝑏 ∈ ℝ⁺ 𝑣 = (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
148	147	ad2antll 761	. . . . . . . . 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)))
149	135, 148	r19.29vva 3062	. . . . . . . 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))))
150	82	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝑋 ≠ ∅)
151	44	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝐷 ∈ (PsMet‘𝑋))
152		metuel 22179	. . . . . . . . 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))))))
153	150, 151, 152	syl2anc 691	. . . . . . . 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))))))
154	97, 149, 153	mpbir2and 959	. . . . . . 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))
155		indir 3834	. . . . . . . . 9 ⊢ ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)) = ((𝑢 ∩ (𝐴 × 𝐴)) ∪ (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴)))
156		incom 3767	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐴) ∩ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) = (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴))
157		disjdif 3992	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐴) ∩ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) = ∅
158	156, 157	eqtr3i 2634	. . . . . . . . . 10 ⊢ (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴)) = ∅
159	158	uneq2i 3726	. . . . . . . . 9 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) ∪ (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴))) = ((𝑢 ∩ (𝐴 × 𝐴)) ∪ ∅)
160		un0 3919	. . . . . . . . 9 ⊢ ((𝑢 ∩ (𝐴 × 𝐴)) ∪ ∅) = (𝑢 ∩ (𝐴 × 𝐴))
161	155, 159, 160	3eqtri 2636	. . . . . . . 8 ⊢ ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)) = (𝑢 ∩ (𝐴 × 𝐴))
162		df-ss 3554	. . . . . . . . 9 ⊢ (𝑢 ⊆ (𝐴 × 𝐴) ↔ (𝑢 ∩ (𝐴 × 𝐴)) = 𝑢)
163	91, 162	sylib 207	. . . . . . . 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)
164	161, 163	syl5req 2657	. . . . . . 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)))
165		ineq1 3769	. . . . . . . . 9 ⊢ (𝑣 = (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) → (𝑣 ∩ (𝐴 × 𝐴)) = ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)))
166	165	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑣 = (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) → (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) ↔ 𝑢 = ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴))))
167	166	rspcev 3282	. . . . . . 7 ⊢ (((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∈ (metUnif‘𝐷) ∧ 𝑢 = ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴))) → ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
168	154, 164, 167	syl2anc 691	. . . . . 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)))
169	89, 168	impbida 873	. . . . 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) ≠ ∅)))
170		eqid 2610	. . . . . . 7 ⊢ (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) = (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴)))
171	170	elrnmpt 5293	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 ∈ ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) ↔ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))))
172	23, 171	ax-mp 5	. . . . 5 ⊢ (𝑢 ∈ ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) ↔ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
173		pweq 4111	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → 𝒫 𝑣 = 𝒫 𝑢)
174	173	ineq2d 3776	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) = (ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢))
175	174	neeq1d 2841	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) ≠ ∅ ↔ (ran (𝑎 ∈ ℝ⁺ ↦ (^◡(𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅))
176	175	elrab 3331	. . . . 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) ≠ ∅))
177	169, 172, 176	3bitr4g 302	. . . 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) ≠ ∅}))
178	177	eqrdv 2608	. . 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) ≠ ∅})
179	19, 178	eqtrd 2644	. 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) ≠ ∅})
180	11, 13, 179	3eqtr4rd 2655	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((metUnif‘𝐷) ↾_t (𝐴 × 𝐴)) = (metUnif‘(𝐷 ↾ (𝐴 × 𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 {crab 2900 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ↦ cmpt 4643 × cxp 5036 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℝ^*cxr 9952 ℝ⁺crp 11708 [,)cico 12048 ↾_t crest 15904 PsMetcpsmet 19551 fBascfbas 19555 filGencfg 19556 metUnifcmetu 19558

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-rp 11709 df-ico 12052 df-rest 15906 df-psmet 19559 df-fbas 19564 df-fg 19565 df-metu 19566

This theorem is referenced by: reust 22977 qqhucn 29364

W3C validator