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Theorem restmetu 24599
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 1135 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → 𝐴 ≠ ∅)
2 psmetres2 24340 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴))
323adant1 1129 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴))
4 oveq2 7439 . . . . . . . 8 (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
54imaeq2d 6080 . . . . . . 7 (𝑎 = 𝑏 → ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
65cbvmptv 5261 . . . . . 6 (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
76rneqi 5951 . . . . 5 ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
87metustfbas 24586 . . . 4 ((𝐴 ≠ ∅ ∧ (𝐷 ↾ (𝐴 × 𝐴)) ∈ (PsMet‘𝐴)) → ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∈ (fBas‘(𝐴 × 𝐴)))
91, 3, 8syl2anc 584 . . 3 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∈ (fBas‘(𝐴 × 𝐴)))
10 fgval 23894 . . 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 24578 . . 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 6920 . . . 4 (metUnif‘𝐷) ∈ V
153elfvexd 6946 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
1615, 15xpexd 7770 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝐴 × 𝐴) ∈ V)
17 restval 17473 . . . 4 (((metUnif‘𝐷) ∈ V ∧ (𝐴 × 𝐴) ∈ V) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))))
1814, 16, 17sylancr 587 . . 3 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))))
19 inss2 4246 . . . . . . . . . . 11 (𝑣 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
20 sseq1 4021 . . . . . . . . . . 11 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → (𝑢 ⊆ (𝐴 × 𝐴) ↔ (𝑣 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)))
2119, 20mpbiri 258 . . . . . . . . . 10 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ⊆ (𝐴 × 𝐴))
22 vex 3482 . . . . . . . . . . 11 𝑢 ∈ V
2322elpw 4609 . . . . . . . . . 10 (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑢 ⊆ (𝐴 × 𝐴))
2421, 23sylibr 234 . . . . . . . . 9 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
2524rexlimivw 3149 . . . . . . . 8 (∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
2625adantl 481 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
27 nfv 1912 . . . . . . . . . . . 12 𝑎(((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
28 nfmpt1 5256 . . . . . . . . . . . . . 14 𝑎(𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
2928nfrn 5966 . . . . . . . . . . . . 13 𝑎ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
3029nfcri 2895 . . . . . . . . . . . 12 𝑎 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
3127, 30nfan 1897 . . . . . . . . . . 11 𝑎((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))))
32 nfv 1912 . . . . . . . . . . 11 𝑎 𝑤𝑣
3331, 32nfan 1897 . . . . . . . . . 10 𝑎(((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))) ∧ 𝑤𝑣)
34 nfmpt1 5256 . . . . . . . . . . . . 13 𝑎(𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
3534nfrn 5966 . . . . . . . . . . . 12 𝑎ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
36 nfcv 2903 . . . . . . . . . . . 12 𝑎𝒫 𝑢
3735, 36nfin 4232 . . . . . . . . . . 11 𝑎(ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢)
38 nfcv 2903 . . . . . . . . . . 11 𝑎
3937, 38nfne 3041 . . . . . . . . . 10 𝑎(ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅
40 simplr 769 . . . . . . . . . . . . 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+)
41 ineq1 4221 . . . . . . . . . . . . . . 15 (𝑤 = (𝐷 “ (0[,)𝑎)) → (𝑤 ∩ (𝐴 × 𝐴)) = ((𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
4241adantl 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)))
43 simp2 1136 . . . . . . . . . . . . . . . 16 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → 𝐷 ∈ (PsMet‘𝑋))
44 psmetf 24332 . . . . . . . . . . . . . . . 16 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
45 ffun 6740 . . . . . . . . . . . . . . . 16 (𝐷:(𝑋 × 𝑋)⟶ℝ* → Fun 𝐷)
46 respreima 7086 . . . . . . . . . . . . . . . 16 (Fun 𝐷 → ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = ((𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
4743, 44, 45, 464syl 19 . . . . . . . . . . . . . . 15 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)) = ((𝐷 “ (0[,)𝑎)) ∩ (𝐴 × 𝐴)))
4847ad6antr 736 . . . . . . . . . . . . . 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)))
4942, 48eqtr4d 2778 . . . . . . . . . . . . 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)))
50 rspe 3247 . . . . . . . . . . . . 13 ((𝑎 ∈ ℝ+ ∧ (𝑤 ∩ (𝐴 × 𝐴)) = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) → ∃𝑎 ∈ ℝ+ (𝑤 ∩ (𝐴 × 𝐴)) = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
5140, 49, 50syl2anc 584 . . . . . . . . . . . 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)))
52 vex 3482 . . . . . . . . . . . . . 14 𝑤 ∈ V
5352inex1 5323 . . . . . . . . . . . . 13 (𝑤 ∩ (𝐴 × 𝐴)) ∈ V
54 eqid 2735 . . . . . . . . . . . . . 14 (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
5554elrnmpt 5972 . . . . . . . . . . . . 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))))
5653, 55ax-mp 5 . . . . . . . . . . . 12 ((𝑤 ∩ (𝐴 × 𝐴)) ∈ ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ (𝑤 ∩ (𝐴 × 𝐴)) = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎)))
5751, 56sylibr 234 . . . . . . . . . . 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))))
58 simpllr 776 . . . . . . . . . . . . 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)
59 ssinss1 4254 . . . . . . . . . . . . 13 (𝑤𝑣 → (𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣)
6058, 59syl 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)
61 inss2 4246 . . . . . . . . . . . . 13 (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
6261a1i 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))
63 pweq 4619 . . . . . . . . . . . . . . . 16 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → 𝒫 𝑢 = 𝒫 (𝑣 ∩ (𝐴 × 𝐴)))
6463eleq2d 2825 . . . . . . . . . . . . . . 15 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ (𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 (𝑣 ∩ (𝐴 × 𝐴))))
6553elpw 4609 . . . . . . . . . . . . . . 15 ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 (𝑣 ∩ (𝐴 × 𝐴)) ↔ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
6664, 65bitrdi 287 . . . . . . . . . . . . . 14 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴))))
67 ssin 4247 . . . . . . . . . . . . . 14 (((𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣 ∧ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)) ↔ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
6866, 67bitr4di 289 . . . . . . . . . . . . 13 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑤 ∩ (𝐴 × 𝐴)) ∈ 𝒫 𝑢 ↔ ((𝑤 ∩ (𝐴 × 𝐴)) ⊆ 𝑣 ∧ (𝑤 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴))))
6968ad5antlr 735 . . . . . . . . . . . 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))))
7060, 62, 69mpbir2and 713 . . . . . . . . . . 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)
71 inelcm 4471 . . . . . . . . . . 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) ≠ ∅)
7257, 70, 71syl2anc 584 . . . . . . . . . 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) ≠ ∅)
73 simplr 769 . . . . . . . . . . 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))))
74 eqid 2735 . . . . . . . . . . . . 13 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
7574elrnmpt 5972 . . . . . . . . . . . 12 (𝑤 ∈ V → (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑎))))
7675elv 3483 . . . . . . . . . . 11 (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑎)))
7773, 76sylib 218 . . . . . . . . . 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)))
7833, 39, 72, 77r19.29af2 3265 . . . . . . . . 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) ≠ ∅)
79 ssn0 4410 . . . . . . . . . . . . . 14 ((𝐴𝑋𝐴 ≠ ∅) → 𝑋 ≠ ∅)
8079ancoms 458 . . . . . . . . . . . . 13 ((𝐴 ≠ ∅ ∧ 𝐴𝑋) → 𝑋 ≠ ∅)
81803adant2 1130 . . . . . . . . . . . 12 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → 𝑋 ≠ ∅)
82 metuel 24593 . . . . . . . . . . . 12 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑣 ∈ (metUnif‘𝐷) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑣)))
8381, 43, 82syl2anc 584 . . . . . . . . . . 11 ((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝑣 ∈ (metUnif‘𝐷) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑣)))
8483simplbda 499 . . . . . . . . . 10 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) → ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑣)
8584adantr 480 . . . . . . . . 9 ((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑣)
8678, 85r19.29a 3160 . . . . . . . 8 ((((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (metUnif‘𝐷)) ∧ 𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)
8786r19.29an 3156 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))) → (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)
8826, 87jca 511 . . . . . 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) ≠ ∅))
89 simprl 771 . . . . . . . . . . 11 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝑢 ∈ 𝒫 (𝐴 × 𝐴))
9089elpwid 4614 . . . . . . . . . 10 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝑢 ⊆ (𝐴 × 𝐴))
91 simpl3 1192 . . . . . . . . . . 11 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝐴𝑋)
92 xpss12 5704 . . . . . . . . . . 11 ((𝐴𝑋𝐴𝑋) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
9391, 91, 92syl2anc 584 . . . . . . . . . 10 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
9490, 93sstrd 4006 . . . . . . . . 9 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝑢 ⊆ (𝑋 × 𝑋))
95 difssd 4147 . . . . . . . . 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))
9694, 95unssd 4202 . . . . . . . 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))
97 simplr 769 . . . . . . . . . . . 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+)
98 eqidd 2736 . . . . . . . . . . . 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)))
994imaeq2d 6080 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)𝑏)))
10099rspceeqv 3645 . . . . . . . . . . . 12 ((𝑏 ∈ ℝ+ ∧ (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑏))) → ∃𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑎)))
10197, 98, 100syl2anc 584 . . . . . . . . . . 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)))
10243ad4antr 732 . . . . . . . . . . . 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))
103 cnvexg 7947 . . . . . . . . . . . 12 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
104 imaexg 7936 . . . . . . . . . . . 12 (𝐷 ∈ V → (𝐷 “ (0[,)𝑏)) ∈ V)
10574elrnmpt 5972 . . . . . . . . . . . 12 ((𝐷 “ (0[,)𝑏)) ∈ V → ((𝐷 “ (0[,)𝑏)) ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)𝑎))))
106102, 103, 104, 1054syl 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))))
107101, 106mpbird 257 . . . . . . . . . 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))))
108 cnvimass 6102 . . . . . . . . . . . . . . . 16 (𝐷 “ (0[,)𝑏)) ⊆ dom 𝐷
109108, 44fssdm 6756 . . . . . . . . . . . . . . 15 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 “ (0[,)𝑏)) ⊆ (𝑋 × 𝑋))
110102, 109syl 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))
111 ssdif0 4372 . . . . . . . . . . . . . 14 ((𝐷 “ (0[,)𝑏)) ⊆ (𝑋 × 𝑋) ↔ ((𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) = ∅)
112110, 111sylib 218 . . . . . . . . . . . . 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)) = ∅)
113 0ss 4406 . . . . . . . . . . . . 13 ∅ ⊆ 𝑢
114112, 113eqsstrdi 4050 . . . . . . . . . . . 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)
115 respreima 7086 . . . . . . . . . . . . . 14 (Fun 𝐷 → ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)) = ((𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴)))
116102, 44, 45, 1154syl 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)))
117 simpr 484 . . . . . . . . . . . . . 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)))
118 simpllr 776 . . . . . . . . . . . . . . 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)
119118elpwid 4614 . . . . . . . . . . . . . 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)
120117, 119eqsstrrd 4035 . . . . . . . . . . . . 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)
121116, 120eqsstrrd 4035 . . . . . . . . . . . 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)
122114, 121unssd 4202 . . . . . . . . . . 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)
123 ssundif 4494 . . . . . . . . . . . 12 ((𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ↔ ((𝐷 “ (0[,)𝑏)) ∖ 𝑢) ⊆ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)))
124 difcom 4495 . . . . . . . . . . . 12 (((𝐷 “ (0[,)𝑏)) ∖ 𝑢) ⊆ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ↔ ((𝐷 “ (0[,)𝑏)) ∖ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ⊆ 𝑢)
125 difdif2 4302 . . . . . . . . . . . . 13 ((𝐷 “ (0[,)𝑏)) ∖ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) = (((𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ∪ ((𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴)))
126125sseq1i 4024 . . . . . . . . . . . 12 (((𝐷 “ (0[,)𝑏)) ∖ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ⊆ 𝑢 ↔ (((𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ∪ ((𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
127123, 124, 1263bitri 297 . . . . . . . . . . 11 ((𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ↔ (((𝐷 “ (0[,)𝑏)) ∖ (𝑋 × 𝑋)) ∪ ((𝐷 “ (0[,)𝑏)) ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
128122, 127sylibr 234 . . . . . . . . . 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))))
129 sseq1 4021 . . . . . . . . . . 11 (𝑤 = (𝐷 “ (0[,)𝑏)) → (𝑤 ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ↔ (𝐷 “ (0[,)𝑏)) ⊆ (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)))))
130129rspcev 3622 . . . . . . . . . 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))))
131107, 128, 130syl2anc 584 . . . . . . . . 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))))
132 elin 3979 . . . . . . . . . . . . . 14 (𝑣 ∈ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ↔ (𝑣 ∈ ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∧ 𝑣 ∈ 𝒫 𝑢))
1336elrnmpt 5972 . . . . . . . . . . . . . . . 16 (𝑣 ∈ V → (𝑣 ∈ ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
134133elv 3483 . . . . . . . . . . . . . . 15 (𝑣 ∈ ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ↔ ∃𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
135134anbi1i 624 . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . . 14 ((∃𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)) ∧ 𝑣 ∈ 𝒫 𝑢) ↔ (𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
137132, 135, 1363bitri 297 . . . . . . . . . . . . 13 (𝑣 ∈ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ↔ (𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
138137exbii 1845 . . . . . . . . . . . 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 4359 . . . . . . . . . . . 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 3069 . . . . . . . . . . . 12 (∃𝑣 ∈ 𝒫 𝑢𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)) ↔ ∃𝑣(𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏))))
141138, 139, 1403bitr4i 303 . . . . . . . . . . 11 ((ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅ ↔ ∃𝑣 ∈ 𝒫 𝑢𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
142141biimpi 216 . . . . . . . . . 10 ((ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅ → ∃𝑣 ∈ 𝒫 𝑢𝑏 ∈ ℝ+ 𝑣 = ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑏)))
143142ad2antll 729 . . . . . . . . 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)))
144131, 143r19.29vva 3214 . . . . . . . 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))))
14581adantr 480 . . . . . . . . 9 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝑋 ≠ ∅)
14643adantr 480 . . . . . . . . 9 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → 𝐷 ∈ (PsMet‘𝑋))
147 metuel 24593 . . . . . . . . 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))))))
148145, 146, 147syl2anc 584 . . . . . . . 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))))))
14996, 144, 148mpbir2and 713 . . . . . . 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))
150 indir 4292 . . . . . . . . 9 ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)) = ((𝑢 ∩ (𝐴 × 𝐴)) ∪ (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴)))
151 disjdifr 4479 . . . . . . . . . 10 (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴)) = ∅
152151uneq2i 4175 . . . . . . . . 9 ((𝑢 ∩ (𝐴 × 𝐴)) ∪ (((𝑋 × 𝑋) ∖ (𝐴 × 𝐴)) ∩ (𝐴 × 𝐴))) = ((𝑢 ∩ (𝐴 × 𝐴)) ∪ ∅)
153 un0 4400 . . . . . . . . 9 ((𝑢 ∩ (𝐴 × 𝐴)) ∪ ∅) = (𝑢 ∩ (𝐴 × 𝐴))
154150, 152, 1533eqtri 2767 . . . . . . . 8 ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)) = (𝑢 ∩ (𝐴 × 𝐴))
155 dfss2 3981 . . . . . . . . 9 (𝑢 ⊆ (𝐴 × 𝐴) ↔ (𝑢 ∩ (𝐴 × 𝐴)) = 𝑢)
15690, 155sylib 218 . . . . . . . 8 (((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) ∧ (𝑢 ∈ 𝒫 (𝐴 × 𝐴) ∧ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅)) → (𝑢 ∩ (𝐴 × 𝐴)) = 𝑢)
157154, 156eqtr2id 2788 . . . . . . 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)))
158 ineq1 4221 . . . . . . . 8 (𝑣 = (𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) → (𝑣 ∩ (𝐴 × 𝐴)) = ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴)))
159158rspceeqv 3645 . . . . . . 7 (((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∈ (metUnif‘𝐷) ∧ 𝑢 = ((𝑢 ∪ ((𝑋 × 𝑋) ∖ (𝐴 × 𝐴))) ∩ (𝐴 × 𝐴))) → ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
160149, 157, 159syl2anc 584 . . . . . 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)))
16188, 160impbida 801 . . . . 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) ≠ ∅)))
162 eqid 2735 . . . . . . 7 (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) = (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴)))
163162elrnmpt 5972 . . . . . 6 (𝑢 ∈ V → (𝑢 ∈ ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) ↔ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴))))
164163elv 3483 . . . . 5 (𝑢 ∈ ran (𝑣 ∈ (metUnif‘𝐷) ↦ (𝑣 ∩ (𝐴 × 𝐴))) ↔ ∃𝑣 ∈ (metUnif‘𝐷)𝑢 = (𝑣 ∩ (𝐴 × 𝐴)))
165 pweq 4619 . . . . . . . 8 (𝑣 = 𝑢 → 𝒫 𝑣 = 𝒫 𝑢)
166165ineq2d 4228 . . . . . . 7 (𝑣 = 𝑢 → (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) = (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢))
167166neeq1d 2998 . . . . . 6 (𝑣 = 𝑢 → ((ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑣) ≠ ∅ ↔ (ran (𝑎 ∈ ℝ+ ↦ ((𝐷 ↾ (𝐴 × 𝐴)) “ (0[,)𝑎))) ∩ 𝒫 𝑢) ≠ ∅))
168167elrab 3695 . . . . 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) ≠ ∅))
169161, 164, 1683bitr4g 314 . . . 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) ≠ ∅}))
170169eqrdv 2733 . . 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) ≠ ∅})
17118, 170eqtrd 2775 . 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) ≠ ∅})
17211, 13, 1713eqtr4rd 2786 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 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wne 2938  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  cmpt 5231   × cxp 5687  ccnv 5688  ran crn 5690  cres 5691  cima 5692  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  0cc0 11153  *cxr 11292  +crp 13032  [,)cico 13386  t crest 17467  PsMetcpsmet 21366  fBascfbas 21370  filGencfg 21371  metUnifcmetu 21373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-rp 13033  df-ico 13390  df-rest 17469  df-psmet 21374  df-fbas 21379  df-fg 21380  df-metu 21381
This theorem is referenced by:  reust  25429  qqhucn  33955
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