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Theorem imasf1oxms 22495
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 2760 . . . . 5 ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
6 eqid 2760 . . . . 5 (dist‘𝑈) = (dist‘𝑈)
7 eqid 2760 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
8 eqid 2760 . . . . . . . 8 ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
97, 8xmsxmet 22462 . . . . . . 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 5298 . . . . . . 7 (𝜑 → (𝑉 × 𝑉) = ((Base‘𝑅) × (Base‘𝑅)))
1211reseq2d 5551 . . . . . 6 (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
132fveq2d 6356 . . . . . 6 (𝜑 → (∞Met‘𝑉) = (∞Met‘(Base‘𝑅)))
1410, 12, 133eltr4d 2854 . . . . 5 (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
151, 2, 3, 4, 5, 6, 14imasf1oxmet 22381 . . . 4 (𝜑 → (dist‘𝑈) ∈ (∞Met‘𝐵))
16 f1ofo 6305 . . . . . . 7 (𝐹:𝑉1-1-onto𝐵𝐹:𝑉onto𝐵)
173, 16syl 17 . . . . . 6 (𝜑𝐹:𝑉onto𝐵)
181, 2, 17, 4imasbas 16374 . . . . 5 (𝜑𝐵 = (Base‘𝑈))
1918fveq2d 6356 . . . 4 (𝜑 → (∞Met‘𝐵) = (∞Met‘(Base‘𝑈)))
2015, 19eleqtrd 2841 . . 3 (𝜑 → (dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)))
21 ssid 3765 . . 3 (Base‘𝑈) ⊆ (Base‘𝑈)
22 xmetres2 22367 . . 3 (((dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)) ∧ (Base‘𝑈) ⊆ (Base‘𝑈)) → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
2320, 21, 22sylancl 697 . 2 (𝜑 → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
24 eqid 2760 . . . 4 (TopOpen‘𝑅) = (TopOpen‘𝑅)
25 eqid 2760 . . . 4 (TopOpen‘𝑈) = (TopOpen‘𝑈)
261, 2, 17, 4, 24, 25imastopn 21725 . . 3 (𝜑 → (TopOpen‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹))
2724, 7, 8xmstopn 22457 . . . . . 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 6356 . . . . 5 (𝜑 → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
3028, 29eqtr4d 2797 . . . 4 (𝜑 → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))
3130oveq1d 6828 . . 3 (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹))
32 blbas 22436 . . . . . 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 22426 . . . . . . 7 (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = 𝑉)
35 f1oeq2 6289 . . . . . . 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 247 . . . . 5 (𝜑𝐹: ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto𝐵)
38 eqid 2760 . . . . . 6 ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
3938tgqtop 21717 . . . . 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 696 . . . 4 (𝜑 → ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
41 eqid 2760 . . . . . . 7 (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
4241mopnval 22444 . . . . . 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 6828 . . . 4 (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹))
45 eqid 2760 . . . . . . 7 (MetOpen‘(dist‘𝑈)) = (MetOpen‘(dist‘𝑈))
4645mopnval 22444 . . . . . 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 22335 . . . . . . . 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 6206 . . . . . . 7 ((dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ* → (dist‘𝑈) Fn ((Base‘𝑈) × (Base‘𝑈)))
51 fnresdm 6161 . . . . . . 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 6356 . . . . 5 (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (MetOpen‘(dist‘𝑈)))
543ad2antrr 764 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝐹:𝑉1-1-onto𝐵)
55 f1of1 6297 . . . . . . . . . . . . . . 15 (𝐹:𝑉1-1-onto𝐵𝐹:𝑉1-1𝐵)
5654, 55syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝐹:𝑉1-1𝐵)
57 cnvimass 5643 . . . . . . . . . . . . . . 15 (𝐹𝑥) ⊆ dom 𝐹
58 f1odm 6302 . . . . . . . . . . . . . . . 16 (𝐹:𝑉1-1-onto𝐵 → dom 𝐹 = 𝑉)
5954, 58syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → dom 𝐹 = 𝑉)
6057, 59syl5sseq 3794 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹𝑥) ⊆ 𝑉)
6114ad2antrr 764 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
62 simprl 811 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑦𝑉)
63 simprr 813 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑟 ∈ ℝ*)
64 blssm 22424 . . . . . . . . . . . . . . 15 ((((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) ∧ 𝑦𝑉𝑟 ∈ ℝ*) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
6561, 62, 63, 64syl3anc 1477 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
66 f1imaeq 6685 . . . . . . . . . . . . . 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 1475 . . . . . . . . . . . . 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 809 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑥𝐵)
70 foimacnv 6315 . . . . . . . . . . . . . . 15 ((𝐹:𝑉onto𝐵𝑥𝐵) → (𝐹 “ (𝐹𝑥)) = 𝑥)
7168, 69, 70syl2anc 696 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹 “ (𝐹𝑥)) = 𝑥)
721ad2antrr 764 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑈 = (𝐹s 𝑅))
732ad2antrr 764 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑉 = (Base‘𝑅))
744ad2antrr 764 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑅 ∈ ∞MetSp)
7572, 73, 54, 74, 5, 6, 61, 62, 63imasf1obl 22494 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)))
7675eqcomd 2766 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟))
7771, 76eqeq12d 2775 . . . . . . . . . . . . 13 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((𝐹 “ (𝐹𝑥)) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) ↔ 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
7867, 77bitr3d 270 . . . . . . . . . . . 12 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
79782rexbidva 3194 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (∃𝑦𝑉𝑟 ∈ ℝ* (𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ ∃𝑦𝑉𝑟 ∈ ℝ* 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
803adantr 472 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝐹:𝑉1-1-onto𝐵)
81 f1ofn 6299 . . . . . . . . . . . 12 (𝐹:𝑉1-1-onto𝐵𝐹 Fn 𝑉)
82 oveq1 6820 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑦) → (𝑧(ball‘(dist‘𝑈))𝑟) = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟))
8382eqeq2d 2770 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑦) → (𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
8483rexbidv 3190 . . . . . . . . . . . . 13 (𝑧 = (𝐹𝑦) → (∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑟 ∈ ℝ* 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
8584rexrn 6524 . . . . . . . . . . . 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 6279 . . . . . . . . . . . . 13 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
8880, 16, 873syl 18 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → ran 𝐹 = 𝐵)
8988rexeqdv 3284 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (∃𝑧 ∈ ran 𝐹𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑧𝐵𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
9079, 86, 893bitr2d 296 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (∃𝑦𝑉𝑟 ∈ ℝ* (𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ ∃𝑧𝐵𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
9114adantr 472 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
92 blrn 22415 . . . . . . . . . . 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 472 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (dist‘𝑈) ∈ (∞Met‘𝐵))
95 blrn 22415 . . . . . . . . . . 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 300 . . . . . . . . 9 ((𝜑𝑥𝐵) → ((𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ↔ 𝑥 ∈ ran (ball‘(dist‘𝑈))))
9897pm5.32da 676 . . . . . . . 8 (𝜑 → ((𝑥𝐵 ∧ (𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) ↔ (𝑥𝐵𝑥 ∈ ran (ball‘(dist‘𝑈)))))
99 f1ofo 6305 . . . . . . . . . 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 21706 . . . . . . . . 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 696 . . . . . . . 8 (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ (𝑥𝐵 ∧ (𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))))
103 blf 22413 . . . . . . . . . . . 12 ((dist‘𝑈) ∈ (∞Met‘𝐵) → (ball‘(dist‘𝑈)):(𝐵 × ℝ*)⟶𝒫 𝐵)
104 frn 6214 . . . . . . . . . . . 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 3743 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥 ∈ 𝒫 𝐵))
107 elpwi 4312 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
108106, 107syl6 35 . . . . . . . . 9 (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥𝐵))
109108pm4.71rd 670 . . . . . . . 8 (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) ↔ (𝑥𝐵𝑥 ∈ ran (ball‘(dist‘𝑈)))))
11098, 102, 1093bitr4d 300 . . . . . . 7 (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ 𝑥 ∈ ran (ball‘(dist‘𝑈))))
111110eqrdv 2758 . . . . . 6 (𝜑 → (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ran (ball‘(dist‘𝑈)))
112111fveq2d 6356 . . . . 5 (𝜑 → (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)) = (topGen‘ran (ball‘(dist‘𝑈))))
11347, 53, 1123eqtr4d 2804 . . . 4 (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
11440, 44, 1133eqtr4d 2804 . . 3 (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
11526, 31, 1143eqtrd 2798 . 2 (𝜑 → (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
116 eqid 2760 . . 3 (Base‘𝑈) = (Base‘𝑈)
117 eqid 2760 . . 3 ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))
11825, 116, 117isxms2 22454 . 2 (𝑈 ∈ ∞MetSp ↔ (((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)) ∧ (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))))))
11923, 115, 118sylanbrc 701 1 (𝜑𝑈 ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wrex 3051  wss 3715  𝒫 cpw 4302   cuni 4588   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267  cres 5268  cima 5269   Fn wfn 6044  wf 6045  1-1wf1 6046  ontowfo 6047  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  *cxr 10265  Basecbs 16059  distcds 16152  TopOpenctopn 16284  topGenctg 16300   qTop cqtop 16365  s cimas 16366  ∞Metcxmt 19933  ballcbl 19935  MetOpencmopn 19938  TopBasesctb 20951  ∞MetSpcxme 22323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-q 11982  df-rp 12026  df-xneg 12139  df-xadd 12140  df-xmul 12141  df-fz 12520  df-fzo 12660  df-seq 12996  df-hash 13312  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-rest 16285  df-topn 16286  df-0g 16304  df-gsum 16305  df-topgen 16306  df-xrs 16364  df-qtop 16369  df-imas 16370  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-mulg 17742  df-cntz 17950  df-cmn 18395  df-psmet 19940  df-xmet 19941  df-bl 19943  df-mopn 19944  df-top 20901  df-topon 20918  df-topsp 20939  df-bases 20952  df-xms 22326
This theorem is referenced by:  imasf1oms  22496  xpsxms  22540
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