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Theorem imasf1oxms 22104
 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 2610 . . . . 5 ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
6 eqid 2610 . . . . 5 (dist‘𝑈) = (dist‘𝑈)
7 eqid 2610 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
8 eqid 2610 . . . . . . . 8 ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
97, 8xmsxmet 22071 . . . . . . 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 5065 . . . . . . 7 (𝜑 → (𝑉 × 𝑉) = ((Base‘𝑅) × (Base‘𝑅)))
1211reseq2d 5317 . . . . . 6 (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
132fveq2d 6107 . . . . . 6 (𝜑 → (∞Met‘𝑉) = (∞Met‘(Base‘𝑅)))
1410, 12, 133eltr4d 2703 . . . . 5 (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
151, 2, 3, 4, 5, 6, 14imasf1oxmet 21990 . . . 4 (𝜑 → (dist‘𝑈) ∈ (∞Met‘𝐵))
16 f1ofo 6057 . . . . . . 7 (𝐹:𝑉1-1-onto𝐵𝐹:𝑉onto𝐵)
173, 16syl 17 . . . . . 6 (𝜑𝐹:𝑉onto𝐵)
181, 2, 17, 4imasbas 15995 . . . . 5 (𝜑𝐵 = (Base‘𝑈))
1918fveq2d 6107 . . . 4 (𝜑 → (∞Met‘𝐵) = (∞Met‘(Base‘𝑈)))
2015, 19eleqtrd 2690 . . 3 (𝜑 → (dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)))
21 ssid 3587 . . 3 (Base‘𝑈) ⊆ (Base‘𝑈)
22 xmetres2 21976 . . 3 (((dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)) ∧ (Base‘𝑈) ⊆ (Base‘𝑈)) → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
2320, 21, 22sylancl 693 . 2 (𝜑 → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
24 eqid 2610 . . . 4 (TopOpen‘𝑅) = (TopOpen‘𝑅)
25 eqid 2610 . . . 4 (TopOpen‘𝑈) = (TopOpen‘𝑈)
261, 2, 17, 4, 24, 25imastopn 21333 . . 3 (𝜑 → (TopOpen‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹))
2724, 7, 8xmstopn 22066 . . . . . 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 6107 . . . . 5 (𝜑 → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
3028, 29eqtr4d 2647 . . . 4 (𝜑 → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))
3130oveq1d 6564 . . 3 (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹))
32 blbas 22045 . . . . . 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 22035 . . . . . . 7 (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = 𝑉)
35 f1oeq2 6041 . . . . . . 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 246 . . . . 5 (𝜑𝐹: ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto𝐵)
38 eqid 2610 . . . . . 6 ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
3938tgqtop 21325 . . . . 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 691 . . . 4 (𝜑 → ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
41 eqid 2610 . . . . . . 7 (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
4241mopnval 22053 . . . . . 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 6564 . . . 4 (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹))
45 eqid 2610 . . . . . . 7 (MetOpen‘(dist‘𝑈)) = (MetOpen‘(dist‘𝑈))
4645mopnval 22053 . . . . . 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 21944 . . . . . . . 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 5958 . . . . . . 7 ((dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ* → (dist‘𝑈) Fn ((Base‘𝑈) × (Base‘𝑈)))
51 fnresdm 5914 . . . . . . 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 6107 . . . . 5 (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (MetOpen‘(dist‘𝑈)))
543ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝐹:𝑉1-1-onto𝐵)
55 f1of1 6049 . . . . . . . . . . . . . . 15 (𝐹:𝑉1-1-onto𝐵𝐹:𝑉1-1𝐵)
5654, 55syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝐹:𝑉1-1𝐵)
57 cnvimass 5404 . . . . . . . . . . . . . . 15 (𝐹𝑥) ⊆ dom 𝐹
58 f1odm 6054 . . . . . . . . . . . . . . . 16 (𝐹:𝑉1-1-onto𝐵 → dom 𝐹 = 𝑉)
5954, 58syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → dom 𝐹 = 𝑉)
6057, 59syl5sseq 3616 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹𝑥) ⊆ 𝑉)
6114ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
62 simprl 790 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑦𝑉)
63 simprr 792 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑟 ∈ ℝ*)
64 blssm 22033 . . . . . . . . . . . . . . 15 ((((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) ∧ 𝑦𝑉𝑟 ∈ ℝ*) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
6561, 62, 63, 64syl3anc 1318 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
66 f1imaeq 6423 . . . . . . . . . . . . . 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 1316 . . . . . . . . . . . . 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 788 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑥𝐵)
70 foimacnv 6067 . . . . . . . . . . . . . . 15 ((𝐹:𝑉onto𝐵𝑥𝐵) → (𝐹 “ (𝐹𝑥)) = 𝑥)
7168, 69, 70syl2anc 691 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹 “ (𝐹𝑥)) = 𝑥)
721ad2antrr 758 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑈 = (𝐹s 𝑅))
732ad2antrr 758 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑉 = (Base‘𝑅))
744ad2antrr 758 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑅 ∈ ∞MetSp)
7572, 73, 54, 74, 5, 6, 61, 62, 63imasf1obl 22103 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)))
7675eqcomd 2616 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟))
7771, 76eqeq12d 2625 . . . . . . . . . . . . 13 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((𝐹 “ (𝐹𝑥)) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) ↔ 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
7867, 77bitr3d 269 . . . . . . . . . . . 12 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
79782rexbidva 3038 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (∃𝑦𝑉𝑟 ∈ ℝ* (𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ ∃𝑦𝑉𝑟 ∈ ℝ* 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
803adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝐹:𝑉1-1-onto𝐵)
81 f1ofn 6051 . . . . . . . . . . . 12 (𝐹:𝑉1-1-onto𝐵𝐹 Fn 𝑉)
82 oveq1 6556 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑦) → (𝑧(ball‘(dist‘𝑈))𝑟) = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟))
8382eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑦) → (𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
8483rexbidv 3034 . . . . . . . . . . . . 13 (𝑧 = (𝐹𝑦) → (∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑟 ∈ ℝ* 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
8584rexrn 6269 . . . . . . . . . . . 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 6031 . . . . . . . . . . . . 13 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
8880, 16, 873syl 18 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → ran 𝐹 = 𝐵)
8988rexeqdv 3122 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (∃𝑧 ∈ ran 𝐹𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑧𝐵𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
9079, 86, 893bitr2d 295 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (∃𝑦𝑉𝑟 ∈ ℝ* (𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ ∃𝑧𝐵𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
9114adantr 480 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
92 blrn 22024 . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (dist‘𝑈) ∈ (∞Met‘𝐵))
95 blrn 22024 . . . . . . . . . . 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 299 . . . . . . . . 9 ((𝜑𝑥𝐵) → ((𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ↔ 𝑥 ∈ ran (ball‘(dist‘𝑈))))
9897pm5.32da 671 . . . . . . . 8 (𝜑 → ((𝑥𝐵 ∧ (𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) ↔ (𝑥𝐵𝑥 ∈ ran (ball‘(dist‘𝑈)))))
99 f1ofo 6057 . . . . . . . . . 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 21314 . . . . . . . . 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 691 . . . . . . . 8 (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ (𝑥𝐵 ∧ (𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))))
103 blf 22022 . . . . . . . . . . . 12 ((dist‘𝑈) ∈ (∞Met‘𝐵) → (ball‘(dist‘𝑈)):(𝐵 × ℝ*)⟶𝒫 𝐵)
104 frn 5966 . . . . . . . . . . . 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 3567 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥 ∈ 𝒫 𝐵))
107 elpwi 4117 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
108106, 107syl6 34 . . . . . . . . 9 (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥𝐵))
109108pm4.71rd 665 . . . . . . . 8 (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) ↔ (𝑥𝐵𝑥 ∈ ran (ball‘(dist‘𝑈)))))
11098, 102, 1093bitr4d 299 . . . . . . 7 (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ 𝑥 ∈ ran (ball‘(dist‘𝑈))))
111110eqrdv 2608 . . . . . 6 (𝜑 → (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ran (ball‘(dist‘𝑈)))
112111fveq2d 6107 . . . . 5 (𝜑 → (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)) = (topGen‘ran (ball‘(dist‘𝑈))))
11347, 53, 1123eqtr4d 2654 . . . 4 (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
11440, 44, 1133eqtr4d 2654 . . 3 (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
11526, 31, 1143eqtrd 2648 . 2 (𝜑 → (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
116 eqid 2610 . . 3 (Base‘𝑈) = (Base‘𝑈)
117 eqid 2610 . . 3 ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))
11825, 116, 117isxms2 22063 . 2 (𝑈 ∈ ∞MetSp ↔ (((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)) ∧ (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))))))
11923, 115, 118sylanbrc 695 1 (𝜑𝑈 ∈ ∞MetSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ℝ*cxr 9952  Basecbs 15695  distcds 15777  TopOpenctopn 15905  topGenctg 15921   qTop cqtop 15986   “s cimas 15987  ∞Metcxmt 19552  ballcbl 19554  MetOpencmopn 19557  TopBasesctb 20520  ∞MetSpcxme 21932 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-inf2 8421  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  ax-pre-sup 9893 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-rmo 2904  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  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-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-xrs 15985  df-qtop 15990  df-imas 15991  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-xms 21935 This theorem is referenced by:  imasf1oms  22105  xpsxms  22149
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