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Theorem imasf1oxms 23096
Description: The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
Hypotheses
Ref Expression
imasf1obl.u (𝜑𝑈 = (𝐹s 𝑅))
imasf1obl.v (𝜑𝑉 = (Base‘𝑅))
imasf1obl.f (𝜑𝐹:𝑉1-1-onto𝐵)
imasf1oxms.r (𝜑𝑅 ∈ ∞MetSp)
Assertion
Ref Expression
imasf1oxms (𝜑𝑈 ∈ ∞MetSp)

Proof of Theorem imasf1oxms
Dummy variables 𝑥 𝑟 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasf1obl.u . . . . 5 (𝜑𝑈 = (𝐹s 𝑅))
2 imasf1obl.v . . . . 5 (𝜑𝑉 = (Base‘𝑅))
3 imasf1obl.f . . . . 5 (𝜑𝐹:𝑉1-1-onto𝐵)
4 imasf1oxms.r . . . . 5 (𝜑𝑅 ∈ ∞MetSp)
5 eqid 2798 . . . . 5 ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
6 eqid 2798 . . . . 5 (dist‘𝑈) = (dist‘𝑈)
7 eqid 2798 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
8 eqid 2798 . . . . . . . 8 ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
97, 8xmsxmet 23063 . . . . . . 7 (𝑅 ∈ ∞MetSp → ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) ∈ (∞Met‘(Base‘𝑅)))
104, 9syl 17 . . . . . 6 (𝜑 → ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) ∈ (∞Met‘(Base‘𝑅)))
112sqxpeqd 5551 . . . . . . 7 (𝜑 → (𝑉 × 𝑉) = ((Base‘𝑅) × (Base‘𝑅)))
1211reseq2d 5818 . . . . . 6 (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
132fveq2d 6649 . . . . . 6 (𝜑 → (∞Met‘𝑉) = (∞Met‘(Base‘𝑅)))
1410, 12, 133eltr4d 2905 . . . . 5 (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
151, 2, 3, 4, 5, 6, 14imasf1oxmet 22982 . . . 4 (𝜑 → (dist‘𝑈) ∈ (∞Met‘𝐵))
16 f1ofo 6597 . . . . . . 7 (𝐹:𝑉1-1-onto𝐵𝐹:𝑉onto𝐵)
173, 16syl 17 . . . . . 6 (𝜑𝐹:𝑉onto𝐵)
181, 2, 17, 4imasbas 16777 . . . . 5 (𝜑𝐵 = (Base‘𝑈))
1918fveq2d 6649 . . . 4 (𝜑 → (∞Met‘𝐵) = (∞Met‘(Base‘𝑈)))
2015, 19eleqtrd 2892 . . 3 (𝜑 → (dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)))
21 ssid 3937 . . 3 (Base‘𝑈) ⊆ (Base‘𝑈)
22 xmetres2 22968 . . 3 (((dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)) ∧ (Base‘𝑈) ⊆ (Base‘𝑈)) → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
2320, 21, 22sylancl 589 . 2 (𝜑 → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
24 eqid 2798 . . . 4 (TopOpen‘𝑅) = (TopOpen‘𝑅)
25 eqid 2798 . . . 4 (TopOpen‘𝑈) = (TopOpen‘𝑈)
261, 2, 17, 4, 24, 25imastopn 22325 . . 3 (𝜑 → (TopOpen‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹))
2724, 7, 8xmstopn 23058 . . . . . 6 (𝑅 ∈ ∞MetSp → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
284, 27syl 17 . . . . 5 (𝜑 → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
2912fveq2d 6649 . . . . 5 (𝜑 → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
3028, 29eqtr4d 2836 . . . 4 (𝜑 → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))
3130oveq1d 7150 . . 3 (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹))
32 blbas 23037 . . . . . 6 (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ∈ TopBases)
3314, 32syl 17 . . . . 5 (𝜑 → ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ∈ TopBases)
34 unirnbl 23027 . . . . . . 7 (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = 𝑉)
35 f1oeq2 6580 . . . . . . 7 ( ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = 𝑉 → (𝐹: ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto𝐵𝐹:𝑉1-1-onto𝐵))
3614, 34, 353syl 18 . . . . . 6 (𝜑 → (𝐹: ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto𝐵𝐹:𝑉1-1-onto𝐵))
373, 36mpbird 260 . . . . 5 (𝜑𝐹: ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto𝐵)
38 eqid 2798 . . . . . 6 ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
3938tgqtop 22317 . . . . 5 ((ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ∈ TopBases ∧ 𝐹: ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto𝐵) → ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
4033, 37, 39syl2anc 587 . . . 4 (𝜑 → ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
41 eqid 2798 . . . . . . 7 (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
4241mopnval 23045 . . . . . 6 (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))))
4314, 42syl 17 . . . . 5 (𝜑 → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))))
4443oveq1d 7150 . . . 4 (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹))
45 eqid 2798 . . . . . . 7 (MetOpen‘(dist‘𝑈)) = (MetOpen‘(dist‘𝑈))
4645mopnval 23045 . . . . . 6 ((dist‘𝑈) ∈ (∞Met‘𝐵) → (MetOpen‘(dist‘𝑈)) = (topGen‘ran (ball‘(dist‘𝑈))))
4715, 46syl 17 . . . . 5 (𝜑 → (MetOpen‘(dist‘𝑈)) = (topGen‘ran (ball‘(dist‘𝑈))))
48 xmetf 22936 . . . . . . . 8 ((dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)) → (dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ*)
4920, 48syl 17 . . . . . . 7 (𝜑 → (dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ*)
50 ffn 6487 . . . . . . 7 ((dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ* → (dist‘𝑈) Fn ((Base‘𝑈) × (Base‘𝑈)))
51 fnresdm 6438 . . . . . . 7 ((dist‘𝑈) Fn ((Base‘𝑈) × (Base‘𝑈)) → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = (dist‘𝑈))
5249, 50, 513syl 18 . . . . . 6 (𝜑 → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = (dist‘𝑈))
5352fveq2d 6649 . . . . 5 (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (MetOpen‘(dist‘𝑈)))
543ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝐹:𝑉1-1-onto𝐵)
55 f1of1 6589 . . . . . . . . . . . . . . 15 (𝐹:𝑉1-1-onto𝐵𝐹:𝑉1-1𝐵)
5654, 55syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝐹:𝑉1-1𝐵)
57 cnvimass 5916 . . . . . . . . . . . . . . 15 (𝐹𝑥) ⊆ dom 𝐹
58 f1odm 6594 . . . . . . . . . . . . . . . 16 (𝐹:𝑉1-1-onto𝐵 → dom 𝐹 = 𝑉)
5954, 58syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → dom 𝐹 = 𝑉)
6057, 59sseqtrid 3967 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹𝑥) ⊆ 𝑉)
6114ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
62 simprl 770 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑦𝑉)
63 simprr 772 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑟 ∈ ℝ*)
64 blssm 23025 . . . . . . . . . . . . . . 15 ((((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) ∧ 𝑦𝑉𝑟 ∈ ℝ*) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
6561, 62, 63, 64syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
66 f1imaeq 7001 . . . . . . . . . . . . . 14 ((𝐹:𝑉1-1𝐵 ∧ ((𝐹𝑥) ⊆ 𝑉 ∧ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)) → ((𝐹 “ (𝐹𝑥)) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) ↔ (𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)))
6756, 60, 65, 66syl12anc 835 . . . . . . . . . . . . 13 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((𝐹 “ (𝐹𝑥)) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) ↔ (𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)))
6854, 16syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝐹:𝑉onto𝐵)
69 simplr 768 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑥𝐵)
70 foimacnv 6607 . . . . . . . . . . . . . . 15 ((𝐹:𝑉onto𝐵𝑥𝐵) → (𝐹 “ (𝐹𝑥)) = 𝑥)
7168, 69, 70syl2anc 587 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹 “ (𝐹𝑥)) = 𝑥)
721ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑈 = (𝐹s 𝑅))
732ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑉 = (Base‘𝑅))
744ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑅 ∈ ∞MetSp)
7572, 73, 54, 74, 5, 6, 61, 62, 63imasf1obl 23095 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)))
7675eqcomd 2804 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟))
7771, 76eqeq12d 2814 . . . . . . . . . . . . 13 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((𝐹 “ (𝐹𝑥)) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) ↔ 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
7867, 77bitr3d 284 . . . . . . . . . . . 12 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
79782rexbidva 3258 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (∃𝑦𝑉𝑟 ∈ ℝ* (𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ ∃𝑦𝑉𝑟 ∈ ℝ* 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
803adantr 484 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝐹:𝑉1-1-onto𝐵)
81 f1ofn 6591 . . . . . . . . . . . 12 (𝐹:𝑉1-1-onto𝐵𝐹 Fn 𝑉)
82 oveq1 7142 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑦) → (𝑧(ball‘(dist‘𝑈))𝑟) = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟))
8382eqeq2d 2809 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑦) → (𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
8483rexbidv 3256 . . . . . . . . . . . . 13 (𝑧 = (𝐹𝑦) → (∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑟 ∈ ℝ* 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
8584rexrn 6830 . . . . . . . . . . . 12 (𝐹 Fn 𝑉 → (∃𝑧 ∈ ran 𝐹𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑦𝑉𝑟 ∈ ℝ* 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
8680, 81, 853syl 18 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (∃𝑧 ∈ ran 𝐹𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑦𝑉𝑟 ∈ ℝ* 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
87 forn 6568 . . . . . . . . . . . . 13 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
8880, 16, 873syl 18 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → ran 𝐹 = 𝐵)
8988rexeqdv 3365 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (∃𝑧 ∈ ran 𝐹𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑧𝐵𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
9079, 86, 893bitr2d 310 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (∃𝑦𝑉𝑟 ∈ ℝ* (𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ ∃𝑧𝐵𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
9114adantr 484 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
92 blrn 23016 . . . . . . . . . . 11 (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ((𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ↔ ∃𝑦𝑉𝑟 ∈ ℝ* (𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)))
9391, 92syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐵) → ((𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ↔ ∃𝑦𝑉𝑟 ∈ ℝ* (𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)))
9415adantr 484 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (dist‘𝑈) ∈ (∞Met‘𝐵))
95 blrn 23016 . . . . . . . . . . 11 ((dist‘𝑈) ∈ (∞Met‘𝐵) → (𝑥 ∈ ran (ball‘(dist‘𝑈)) ↔ ∃𝑧𝐵𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
9694, 95syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (𝑥 ∈ ran (ball‘(dist‘𝑈)) ↔ ∃𝑧𝐵𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
9790, 93, 963bitr4d 314 . . . . . . . . 9 ((𝜑𝑥𝐵) → ((𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ↔ 𝑥 ∈ ran (ball‘(dist‘𝑈))))
9897pm5.32da 582 . . . . . . . 8 (𝜑 → ((𝑥𝐵 ∧ (𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) ↔ (𝑥𝐵𝑥 ∈ ran (ball‘(dist‘𝑈)))))
99 f1ofo 6597 . . . . . . . . . 10 (𝐹: ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto𝐵𝐹: ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–onto𝐵)
10037, 99syl 17 . . . . . . . . 9 (𝜑𝐹: ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–onto𝐵)
10138elqtop2 22306 . . . . . . . . 9 ((ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ∈ TopBases ∧ 𝐹: ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–onto𝐵) → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ (𝑥𝐵 ∧ (𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))))
10233, 100, 101syl2anc 587 . . . . . . . 8 (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ (𝑥𝐵 ∧ (𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))))
103 blf 23014 . . . . . . . . . . . 12 ((dist‘𝑈) ∈ (∞Met‘𝐵) → (ball‘(dist‘𝑈)):(𝐵 × ℝ*)⟶𝒫 𝐵)
104 frn 6493 . . . . . . . . . . . 12 ((ball‘(dist‘𝑈)):(𝐵 × ℝ*)⟶𝒫 𝐵 → ran (ball‘(dist‘𝑈)) ⊆ 𝒫 𝐵)
10515, 103, 1043syl 18 . . . . . . . . . . 11 (𝜑 → ran (ball‘(dist‘𝑈)) ⊆ 𝒫 𝐵)
106105sseld 3914 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥 ∈ 𝒫 𝐵))
107 elpwi 4506 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
108106, 107syl6 35 . . . . . . . . 9 (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥𝐵))
109108pm4.71rd 566 . . . . . . . 8 (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) ↔ (𝑥𝐵𝑥 ∈ ran (ball‘(dist‘𝑈)))))
11098, 102, 1093bitr4d 314 . . . . . . 7 (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ 𝑥 ∈ ran (ball‘(dist‘𝑈))))
111110eqrdv 2796 . . . . . 6 (𝜑 → (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ran (ball‘(dist‘𝑈)))
112111fveq2d 6649 . . . . 5 (𝜑 → (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)) = (topGen‘ran (ball‘(dist‘𝑈))))
11347, 53, 1123eqtr4d 2843 . . . 4 (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
11440, 44, 1133eqtr4d 2843 . . 3 (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
11526, 31, 1143eqtrd 2837 . 2 (𝜑 → (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
116 eqid 2798 . . 3 (Base‘𝑈) = (Base‘𝑈)
117 eqid 2798 . . 3 ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))
11825, 116, 117isxms2 23055 . 2 (𝑈 ∈ ∞MetSp ↔ (((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)) ∧ (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))))))
11923, 115, 118sylanbrc 586 1 (𝜑𝑈 ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wrex 3107  wss 3881  𝒫 cpw 4497   cuni 4800   × cxp 5517  ccnv 5518  dom cdm 5519  ran crn 5520  cres 5521  cima 5522   Fn wfn 6319  wf 6320  1-1wf1 6321  ontowfo 6322  1-1-ontowf1o 6323  cfv 6324  (class class class)co 7135  *cxr 10663  Basecbs 16475  distcds 16566  TopOpenctopn 16687  topGenctg 16703   qTop cqtop 16768  s cimas 16769  ∞Metcxmet 20076  ballcbl 20078  MetOpencmopn 20081  TopBasesctb 21550  ∞MetSpcxms 22924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-sup 8890  df-inf 8891  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-fz 12886  df-fzo 13029  df-seq 13365  df-hash 13687  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-rest 16688  df-topn 16689  df-0g 16707  df-gsum 16708  df-topgen 16709  df-xrs 16767  df-qtop 16772  df-imas 16773  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-mulg 18217  df-cntz 18439  df-cmn 18900  df-psmet 20083  df-xmet 20084  df-bl 20086  df-mopn 20087  df-top 21499  df-topon 21516  df-topsp 21538  df-bases 21551  df-xms 22927
This theorem is referenced by:  imasf1oms  23097  xpsxms  23141
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