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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) ∧ 𝐴𝑋) → ((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.
StepHypRef Expression
1 simp1 1054 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → 𝐴 ≠ ∅)
2 psmetres2 21929 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴))
323adant1 1072 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴))
4 oveq2 6557 . . . . . . . 8 (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
54imaeq2d 5385 . . . . . . 7 (𝑎 = 𝑏 → ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
65cbvmptv 4678 . . . . . 6 (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
76rneqi 5273 . . . . 5 ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
87metustfbas 22172 . . . 4 ((𝐴 ≠ ∅ ∧ (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴)) → ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∈ (fBas‘(𝐴 × 𝐴)))
91, 3, 8syl2anc 691 . . 3 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → 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) ≠ ∅})
119, 10syl 17 . 2 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → ((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)))))
133, 12syl 17 . 2 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (metUnif‘(𝐷 ↾ (𝐴 × 𝐴))) = ((𝐴 × 𝐴)filGenran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))))
14 fvex 6113 . . . 4 (metUnif‘𝐷) ∈ V
153elfvexd 6132 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
16 xpexg 6858 . . . . 5 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 × 𝐴) ∈ V)
1715, 15, 16syl2anc 691 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (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))))
1914, 17, 18sylancr 694 . . 3 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → ((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)))
2220, 21mpbiri 247 . . . . . . . . . 10 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ⊆ (𝐴 × 𝐴))
23 vex 3176 . . . . . . . . . . 11 𝑢 ∈ V
2423elpw 4114 . . . . . . . . . 10 (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑢 ⊆ (𝐴 × 𝐴))
2522, 24sylibr 223 . . . . . . . . 9 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
2625rexlimivw 3011 . . . . . . . 8 (∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
2726adantl 481 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
28 nfv 1830 . . . . . . . . . . . 12 𝑎(((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
29 nfmpt1 4675 . . . . . . . . . . . . . 14 𝑎(𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
3029nfrn 5289 . . . . . . . . . . . . 13 𝑎ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
3130nfcri 2745 . . . . . . . . . . . 12 𝑎 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
3228, 31nfan 1816 . . . . . . . . . . 11 𝑎((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))))
33 nfv 1830 . . . . . . . . . . 11 𝑎 𝑤𝑣
3432, 33nfan 1816 . . . . . . . . . 10 𝑎(((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣)
35 nfmpt1 4675 . . . . . . . . . . . . 13 𝑎(𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
3635nfrn 5289 . . . . . . . . . . . 12 𝑎ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
37 nfcv 2751 . . . . . . . . . . . 12 𝑎𝒫 𝑢
3836, 37nfin 3782 . . . . . . . . . . 11 𝑎(ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢)
39 nfcv 2751 . . . . . . . . . . 11 𝑎
4038, 39nfne 2882 . . . . . . . . . 10 𝑎(ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅
41 simplr 788 . . . . . . . . . . . . 13 ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (𝐷 “ (0[,)𝑎))) → 𝑎 ∈ ℝ+)
42 ineq1 3769 . . . . . . . . . . . . . . 15 (𝑤 = (𝐷 “ (0[,)𝑎)) → (𝑤 ∩ (𝐴 × 𝐴)) = ((𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
4342adantl 481 . . . . . . . . . . . . . 14 ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (𝐷 “ (0[,)𝑎))) → (𝑤 ∩ (𝐴 × 𝐴)) = ((𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
44 simp2 1055 . . . . . . . . . . . . . . . 16 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → 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)))
4844, 45, 46, 474syl 19 . . . . . . . . . . . . . . 15 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = ((𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
4948ad6antr 768 . . . . . . . . . . . . . 14 ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (𝐷 “ (0[,)𝑎))) → ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = ((𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
5043, 49eqtr4d 2647 . . . . . . . . . . . . 13 ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) ∧ 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)))
5241, 50, 51syl2anc 691 . . . . . . . . . . . 12 ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (𝐷 “ (0[,)𝑎))) → ∃𝑎 ∈ ℝ+ (𝑤 ∩ (𝐴 × 𝐴)) = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
53 vex 3176 . . . . . . . . . . . . . 14 𝑤 ∈ V
5453inex1 4727 . . . . . . . . . . . . 13 (𝑤 ∩ (𝐴 × 𝐴)) ∈ V
55 eqid 2610 . . . . . . . . . . . . . 14 (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
5655elrnmpt 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))))
5754, 56ax-mp 5 . . . . . . . . . . . 12 ((𝑤 ∩ (𝐴 × 𝐴)) ∈ ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ (𝑤 ∩ (𝐴 × 𝐴)) = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
5852, 57sylibr 223 . . . . . . . . . . 11 ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) ∧ 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) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (𝐷 “ (0[,)𝑎))) → 𝑤𝑣)
60 ssinss1 3803 . . . . . . . . . . . . 13 (𝑤𝑣 → (𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣)
6159, 60syl 17 . . . . . . . . . . . 12 ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (𝐷 “ (0[,)𝑎))) → (𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣)
62 inss2 3796 . . . . . . . . . . . . 13 (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
6362a1i 11 . . . . . . . . . . . 12 ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) ∧ 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)))
6564eleq2d 2673 . . . . . . . . . . . . . . 15 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ (𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 (𝑣 ∩ (𝐴 × 𝐴))))
6654elpw 4114 . . . . . . . . . . . . . . 15 ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 (𝑣 ∩ (𝐴 × 𝐴)) ↔ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
6765, 66syl6bb 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)))
6967, 68syl6bbr 277 . . . . . . . . . . . . 13 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ ((𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣 ∧ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴))))
7069ad5antlr 767 . . . . . . . . . . . 12 ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (𝐷 “ (0[,)𝑎))) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ ((𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣 ∧ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴))))
7161, 63, 70mpbir2and 959 . . . . . . . . . . 11 ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) ∧ 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) ≠ ∅)
7358, 71, 72syl2anc 691 . . . . . . . . . 10 ((((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) ∧ 𝑎 ∈ ℝ+) ∧ 𝑤 = (𝐷 “ (0[,)𝑎))) → (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)
74 simplr 788 . . . . . . . . . . 11 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) → 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))))
75 eqid 2610 . . . . . . . . . . . . 13 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
7675elrnmpt 5293 . . . . . . . . . . . 12 (𝑤 ∈ V → (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑎))))
7753, 76ax-mp 5 . . . . . . . . . . 11 (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑎)))
7874, 77sylib 207 . . . . . . . . . 10 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) → ∃𝑎 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑎)))
7934, 40, 73, 78r19.29af2 3057 . . . . . . . . 9 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣) → (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)
80 ssn0 3928 . . . . . . . . . . . . . 14 ((𝐴𝑋𝐴 ≠ ∅) → 𝑋 ≠ ∅)
8180ancoms 468 . . . . . . . . . . . . 13 ((𝐴 ≠ ∅ ∧ 𝐴𝑋) → 𝑋 ≠ ∅)
82813adant2 1073 . . . . . . . . . . . 12 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → 𝑋 ≠ ∅)
83 metuel 22179 . . . . . . . . . . . 12 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑣 ∈ (metUnif‘𝐷) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑣)))
8482, 44, 83syl2anc 691 . . . . . . . . . . 11 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝑣 ∈ (metUnif‘𝐷) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑣)))
8584simplbda 652 . . . . . . . . . 10 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) → ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑣)
8685adantr 480 . . . . . . . . 9 ((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑣)
8779, 86r19.29a 3060 . . . . . . . 8 ((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)
8887r19.29an 3059 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)
8927, 88jca 553 . . . . . 6 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ ∃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) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
9190elpwid 4118 . . . . . . . . . 10 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝐴𝑋)
93 xpss12 5148 . . . . . . . . . . 11 ((𝐴𝑋𝐴𝑋) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
9492, 92, 93syl2anc 691 . . . . . . . . . 10 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
9591, 94sstrd 3578 . . . . . . . . 9 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ⊆ (𝑋 × 𝑋))
9795, 96unssd 3751 . . . . . . . 8 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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) ∧ 𝐴𝑋) ∧ (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) ∧ 𝐴𝑋) ∧ (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)))
1004imaeq2d 5385 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)𝑏)))
101100eqeq2d 2620 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → ((𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑎)) ↔ (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑏))))
102101rspcev 3282 . . . . . . . . . . . 12 ((𝑏 ∈ ℝ+ ∧ (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑏))) → ∃𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑎)))
10398, 99, 102syl2anc 691 . . . . . . . . . . 11 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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)))
10444ad4antr 764 . . . . . . . . . . . 12 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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)
10775elrnmpt 5293 . . . . . . . . . . . 12 ((𝐷 “ (0[,)𝑏)) ∈ V → ((𝐷 “ (0[,)𝑏)) ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑎))))
108104, 105, 106, 1074syl 19 . . . . . . . . . . 11 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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))))
109103, 108mpbird 246 . . . . . . . . . 10 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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))
11245, 111syl 17 . . . . . . . . . . . . . . . 16 (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
113110, 112syl5sseq 3616 . . . . . . . . . . . . . . 15 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 “ (0[,)𝑏)) ⊆ (𝑋 × 𝑋))
114104, 113syl 17 . . . . . . . . . . . . . 14 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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)) = ∅)
116114, 115sylib 207 . . . . . . . . . . . . 13 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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
118116, 117syl6eqss 3618 . . . . . . . . . . . 12 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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)))
120104, 45, 46, 1194syl 19 . . . . . . . . . . . . 13 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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) ∧ 𝐴𝑋) ∧ (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) ∧ 𝐴𝑋) ∧ (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)
123122elpwid 4118 . . . . . . . . . . . . . 14 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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))) → 𝑣𝑢)
124121, 123eqsstr3d 3603 . . . . . . . . . . . . 13 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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)
125120, 124eqsstr3d 3603 . . . . . . . . . . . 12 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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)
126118, 125unssd 3751 . . . . . . . . . . 11 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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)))
130129sseq1i 3592 . . . . . . . . . . . 12 (((𝐷 “ (0[,)𝑏)) ∖ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ⊆ 𝑢 ↔ (((𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ∪ ((𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
131127, 128, 1303bitri 285 . . . . . . . . . . 11 ((𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ↔ (((𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ∪ ((𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
132126, 131sylibr 223 . . . . . . . . . 10 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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)))))
134133rspcev 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))))
135109, 132, 134syl2anc 691 . . . . . . . . 9 ((((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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
1386elrnmpt 5293 . . . . . . . . . . . . . . . 16 (𝑣 ∈ V → (𝑣 ∈ ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
139137, 138ax-mp 5 . . . . . . . . . . . . . . 15 (𝑣 ∈ ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
140139anbi1i 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))))
142136, 140, 1413bitri 285 . . . . . . . . . . . . 13 (𝑣 ∈ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ↔ (𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
143142exbii 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 𝑢𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)) ↔ ∃𝑣(𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
146143, 144, 1453bitr4i 291 . . . . . . . . . . 11 ((ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅ ↔ ∃𝑣 ∈ 𝒫 𝑢𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
147146biimpi 205 . . . . . . . . . 10 ((ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅ → ∃𝑣 ∈ 𝒫 𝑢𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
148147ad2antll 761 . . . . . . . . 9 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → ∃𝑣 ∈ 𝒫 𝑢𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
149135, 148r19.29vva 3062 . . . . . . . 8 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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))))
15082adantr 480 . . . . . . . . 9 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝑋 ≠ ∅)
15144adantr 480 . . . . . . . . 9 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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))))))
153150, 151, 152syl2anc 691 . . . . . . . 8 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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))))))
15497, 149, 153mpbir2and 959 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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))) = ∅
158156, 157eqtr3i 2634 . . . . . . . . . 10 (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴)) = ∅
159158uneq2i 3726 . . . . . . . . 9 ((𝑢 ∩ (𝐴 × 𝐴)) ∪ (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴))) = ((𝑢 ∩ (𝐴 × 𝐴)) ∪ ∅)
160 un0 3919 . . . . . . . . 9 ((𝑢 ∩ (𝐴 × 𝐴)) ∪ ∅) = (𝑢 ∩ (𝐴 × 𝐴))
161155, 159, 1603eqtri 2636 . . . . . . . 8 ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)) = (𝑢 ∩ (𝐴 × 𝐴))
162 df-ss 3554 . . . . . . . . 9 (𝑢 ⊆ (𝐴 × 𝐴) ↔ (𝑢 ∩ (𝐴 × 𝐴)) = 𝑢)
16391, 162sylib 207 . . . . . . . 8 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → (𝑢 ∩ (𝐴 × 𝐴)) = 𝑢)
164161, 163syl5req 2657 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (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)))
166165eqeq2d 2620 . . . . . . . 8 (𝑣 = (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) → (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) ↔ 𝑢 = ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴))))
167166rspcev 3282 . . . . . . 7 (((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∈ (metUnif‘𝐷) ∧ 𝑢 = ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴))) → ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
168154, 164, 167syl2anc 691 . . . . . 6 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
16989, 168impbida 873 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (∃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)))
171170elrnmpt 5293 . . . . . 6 (𝑢 ∈ V → (𝑢 ∈ ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) ↔ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))))
17223, 171ax-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)
174173ineq2d 3776 . . . . . . 7 (𝑣 = 𝑢 → (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) = (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢))
175174neeq1d 2841 . . . . . 6 (𝑣 = 𝑢 → ((ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) ≠ ∅ ↔ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅))
176175elrab 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) ≠ ∅))
177169, 172, 1763bitr4g 302 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝑢 ∈ ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) ↔ 𝑢 ∈ {𝑣 ∈ 𝒫 (𝐴 × 𝐴) ∣ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) ≠ ∅}))
178177eqrdv 2608 . . 3 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) = {𝑣 ∈ 𝒫 (𝐴 × 𝐴) ∣ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) ≠ ∅})
17919, 178eqtrd 2644 . 2 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = {𝑣 ∈ 𝒫 (𝐴 × 𝐴) ∣ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) ≠ ∅})
18011, 13, 1793eqtr4rd 2655 1 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → ((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
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