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Theorem imasf1oxms 22511
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 2813 . . . . 5 ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
6 eqid 2813 . . . . 5 (dist‘𝑈) = (dist‘𝑈)
7 eqid 2813 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
8 eqid 2813 . . . . . . . 8 ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
97, 8xmsxmet 22478 . . . . . . 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 5349 . . . . . . 7 (𝜑 → (𝑉 × 𝑉) = ((Base‘𝑅) × (Base‘𝑅)))
1211reseq2d 5604 . . . . . 6 (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
132fveq2d 6415 . . . . . 6 (𝜑 → (∞Met‘𝑉) = (∞Met‘(Base‘𝑅)))
1410, 12, 133eltr4d 2907 . . . . 5 (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
151, 2, 3, 4, 5, 6, 14imasf1oxmet 22397 . . . 4 (𝜑 → (dist‘𝑈) ∈ (∞Met‘𝐵))
16 f1ofo 6363 . . . . . . 7 (𝐹:𝑉1-1-onto𝐵𝐹:𝑉onto𝐵)
173, 16syl 17 . . . . . 6 (𝜑𝐹:𝑉onto𝐵)
181, 2, 17, 4imasbas 16380 . . . . 5 (𝜑𝐵 = (Base‘𝑈))
1918fveq2d 6415 . . . 4 (𝜑 → (∞Met‘𝐵) = (∞Met‘(Base‘𝑈)))
2015, 19eleqtrd 2894 . . 3 (𝜑 → (dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)))
21 ssid 3827 . . 3 (Base‘𝑈) ⊆ (Base‘𝑈)
22 xmetres2 22383 . . 3 (((dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)) ∧ (Base‘𝑈) ⊆ (Base‘𝑈)) → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
2320, 21, 22sylancl 576 . 2 (𝜑 → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
24 eqid 2813 . . . 4 (TopOpen‘𝑅) = (TopOpen‘𝑅)
25 eqid 2813 . . . 4 (TopOpen‘𝑈) = (TopOpen‘𝑈)
261, 2, 17, 4, 24, 25imastopn 21741 . . 3 (𝜑 → (TopOpen‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹))
2724, 7, 8xmstopn 22473 . . . . . 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 6415 . . . . 5 (𝜑 → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
3028, 29eqtr4d 2850 . . . 4 (𝜑 → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))
3130oveq1d 6892 . . 3 (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹))
32 blbas 22452 . . . . . 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 22442 . . . . . . 7 (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = 𝑉)
35 f1oeq2 6347 . . . . . . 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 248 . . . . 5 (𝜑𝐹: ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto𝐵)
38 eqid 2813 . . . . . 6 ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
3938tgqtop 21733 . . . . 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 575 . . . 4 (𝜑 → ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
41 eqid 2813 . . . . . . 7 (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
4241mopnval 22460 . . . . . 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 6892 . . . 4 (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹))
45 eqid 2813 . . . . . . 7 (MetOpen‘(dist‘𝑈)) = (MetOpen‘(dist‘𝑈))
4645mopnval 22460 . . . . . 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 22351 . . . . . . . 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 6259 . . . . . . 7 ((dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ* → (dist‘𝑈) Fn ((Base‘𝑈) × (Base‘𝑈)))
51 fnresdm 6214 . . . . . . 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 6415 . . . . 5 (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (MetOpen‘(dist‘𝑈)))
543ad2antrr 708 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝐹:𝑉1-1-onto𝐵)
55 f1of1 6355 . . . . . . . . . . . . . . 15 (𝐹:𝑉1-1-onto𝐵𝐹:𝑉1-1𝐵)
5654, 55syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝐹:𝑉1-1𝐵)
57 cnvimass 5702 . . . . . . . . . . . . . . 15 (𝐹𝑥) ⊆ dom 𝐹
58 f1odm 6360 . . . . . . . . . . . . . . . 16 (𝐹:𝑉1-1-onto𝐵 → dom 𝐹 = 𝑉)
5954, 58syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → dom 𝐹 = 𝑉)
6057, 59syl5sseq 3857 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹𝑥) ⊆ 𝑉)
6114ad2antrr 708 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
62 simprl 778 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑦𝑉)
63 simprr 780 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑟 ∈ ℝ*)
64 blssm 22440 . . . . . . . . . . . . . . 15 ((((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) ∧ 𝑦𝑉𝑟 ∈ ℝ*) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
6561, 62, 63, 64syl3anc 1483 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
66 f1imaeq 6749 . . . . . . . . . . . . . 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 856 . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑥𝐵)
70 foimacnv 6373 . . . . . . . . . . . . . . 15 ((𝐹:𝑉onto𝐵𝑥𝐵) → (𝐹 “ (𝐹𝑥)) = 𝑥)
7168, 69, 70syl2anc 575 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹 “ (𝐹𝑥)) = 𝑥)
721ad2antrr 708 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑈 = (𝐹s 𝑅))
732ad2antrr 708 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑉 = (Base‘𝑅))
744ad2antrr 708 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑅 ∈ ∞MetSp)
7572, 73, 54, 74, 5, 6, 61, 62, 63imasf1obl 22510 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)))
7675eqcomd 2819 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟))
7771, 76eqeq12d 2828 . . . . . . . . . . . . 13 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((𝐹 “ (𝐹𝑥)) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) ↔ 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
7867, 77bitr3d 272 . . . . . . . . . . . 12 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
79782rexbidva 3251 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (∃𝑦𝑉𝑟 ∈ ℝ* (𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ ∃𝑦𝑉𝑟 ∈ ℝ* 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
803adantr 468 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝐹:𝑉1-1-onto𝐵)
81 f1ofn 6357 . . . . . . . . . . . 12 (𝐹:𝑉1-1-onto𝐵𝐹 Fn 𝑉)
82 oveq1 6884 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑦) → (𝑧(ball‘(dist‘𝑈))𝑟) = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟))
8382eqeq2d 2823 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑦) → (𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
8483rexbidv 3247 . . . . . . . . . . . . 13 (𝑧 = (𝐹𝑦) → (∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑟 ∈ ℝ* 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
8584rexrn 6586 . . . . . . . . . . . 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 6337 . . . . . . . . . . . . 13 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
8880, 16, 873syl 18 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → ran 𝐹 = 𝐵)
8988rexeqdv 3341 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (∃𝑧 ∈ ran 𝐹𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑧𝐵𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
9079, 86, 893bitr2d 298 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (∃𝑦𝑉𝑟 ∈ ℝ* (𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ ∃𝑧𝐵𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
9114adantr 468 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
92 blrn 22431 . . . . . . . . . . 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 468 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (dist‘𝑈) ∈ (∞Met‘𝐵))
95 blrn 22431 . . . . . . . . . . 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 302 . . . . . . . . 9 ((𝜑𝑥𝐵) → ((𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ↔ 𝑥 ∈ ran (ball‘(dist‘𝑈))))
9897pm5.32da 570 . . . . . . . 8 (𝜑 → ((𝑥𝐵 ∧ (𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) ↔ (𝑥𝐵𝑥 ∈ ran (ball‘(dist‘𝑈)))))
99 f1ofo 6363 . . . . . . . . . 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 21722 . . . . . . . . 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 575 . . . . . . . 8 (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ (𝑥𝐵 ∧ (𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))))
103 blf 22429 . . . . . . . . . . . 12 ((dist‘𝑈) ∈ (∞Met‘𝐵) → (ball‘(dist‘𝑈)):(𝐵 × ℝ*)⟶𝒫 𝐵)
104 frn 6265 . . . . . . . . . . . 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 3804 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥 ∈ 𝒫 𝐵))
107 elpwi 4368 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
108106, 107syl6 35 . . . . . . . . 9 (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥𝐵))
109108pm4.71rd 554 . . . . . . . 8 (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) ↔ (𝑥𝐵𝑥 ∈ ran (ball‘(dist‘𝑈)))))
11098, 102, 1093bitr4d 302 . . . . . . 7 (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ 𝑥 ∈ ran (ball‘(dist‘𝑈))))
111110eqrdv 2811 . . . . . 6 (𝜑 → (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ran (ball‘(dist‘𝑈)))
112111fveq2d 6415 . . . . 5 (𝜑 → (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)) = (topGen‘ran (ball‘(dist‘𝑈))))
11347, 53, 1123eqtr4d 2857 . . . 4 (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
11440, 44, 1133eqtr4d 2857 . . 3 (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
11526, 31, 1143eqtrd 2851 . 2 (𝜑 → (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
116 eqid 2813 . . 3 (Base‘𝑈) = (Base‘𝑈)
117 eqid 2813 . . 3 ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))
11825, 116, 117isxms2 22470 . 2 (𝑈 ∈ ∞MetSp ↔ (((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)) ∧ (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))))))
11923, 115, 118sylanbrc 574 1 (𝜑𝑈 ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  wrex 3104  wss 3776  𝒫 cpw 4358   cuni 4637   × cxp 5316  ccnv 5317  dom cdm 5318  ran crn 5319  cres 5320  cima 5321   Fn wfn 6099  wf 6100  1-1wf1 6101  ontowfo 6102  1-1-ontowf1o 6103  cfv 6104  (class class class)co 6877  *cxr 10361  Basecbs 16071  distcds 16165  TopOpenctopn 16290  topGenctg 16306   qTop cqtop 16371  s cimas 16372  ∞Metcxmt 19942  ballcbl 19944  MetOpencmopn 19947  TopBasesctb 20967  ∞MetSpcxme 22339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-inf2 8788  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-iin 4722  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-se 5278  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-isom 6113  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-of 7130  df-om 7299  df-1st 7401  df-2nd 7402  df-supp 7533  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-fsupp 8518  df-sup 8590  df-inf 8591  df-oi 8657  df-card 9051  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-div 10973  df-nn 11309  df-2 11367  df-3 11368  df-4 11369  df-5 11370  df-6 11371  df-7 11372  df-8 11373  df-9 11374  df-n0 11563  df-z 11647  df-dec 11763  df-uz 11908  df-q 12011  df-rp 12050  df-xneg 12165  df-xadd 12166  df-xmul 12167  df-fz 12553  df-fzo 12693  df-seq 13028  df-hash 13341  df-struct 16073  df-ndx 16074  df-slot 16075  df-base 16077  df-sets 16078  df-ress 16079  df-plusg 16169  df-mulr 16170  df-sca 16172  df-vsca 16173  df-ip 16174  df-tset 16175  df-ple 16176  df-ds 16178  df-rest 16291  df-topn 16292  df-0g 16310  df-gsum 16311  df-topgen 16312  df-xrs 16370  df-qtop 16375  df-imas 16376  df-mre 16454  df-mrc 16455  df-acs 16457  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-submnd 17544  df-mulg 17749  df-cntz 17954  df-cmn 18399  df-psmet 19949  df-xmet 19950  df-bl 19952  df-mopn 19953  df-top 20916  df-topon 20933  df-topsp 20955  df-bases 20968  df-xms 22342
This theorem is referenced by:  imasf1oms  22512  xpsxms  22556
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