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Theorem imasf1oxms 22204
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 2621 . . . . 5 ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
6 eqid 2621 . . . . 5 (dist‘𝑈) = (dist‘𝑈)
7 eqid 2621 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
8 eqid 2621 . . . . . . . 8 ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
97, 8xmsxmet 22171 . . . . . . 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 5101 . . . . . . 7 (𝜑 → (𝑉 × 𝑉) = ((Base‘𝑅) × (Base‘𝑅)))
1211reseq2d 5356 . . . . . 6 (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
132fveq2d 6152 . . . . . 6 (𝜑 → (∞Met‘𝑉) = (∞Met‘(Base‘𝑅)))
1410, 12, 133eltr4d 2713 . . . . 5 (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
151, 2, 3, 4, 5, 6, 14imasf1oxmet 22090 . . . 4 (𝜑 → (dist‘𝑈) ∈ (∞Met‘𝐵))
16 f1ofo 6101 . . . . . . 7 (𝐹:𝑉1-1-onto𝐵𝐹:𝑉onto𝐵)
173, 16syl 17 . . . . . 6 (𝜑𝐹:𝑉onto𝐵)
181, 2, 17, 4imasbas 16093 . . . . 5 (𝜑𝐵 = (Base‘𝑈))
1918fveq2d 6152 . . . 4 (𝜑 → (∞Met‘𝐵) = (∞Met‘(Base‘𝑈)))
2015, 19eleqtrd 2700 . . 3 (𝜑 → (dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)))
21 ssid 3603 . . 3 (Base‘𝑈) ⊆ (Base‘𝑈)
22 xmetres2 22076 . . 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 2621 . . . 4 (TopOpen‘𝑅) = (TopOpen‘𝑅)
25 eqid 2621 . . . 4 (TopOpen‘𝑈) = (TopOpen‘𝑈)
261, 2, 17, 4, 24, 25imastopn 21433 . . 3 (𝜑 → (TopOpen‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹))
2724, 7, 8xmstopn 22166 . . . . . 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 6152 . . . . 5 (𝜑 → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
3028, 29eqtr4d 2658 . . . 4 (𝜑 → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))
3130oveq1d 6619 . . 3 (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹))
32 blbas 22145 . . . . . 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 22135 . . . . . . 7 (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = 𝑉)
35 f1oeq2 6085 . . . . . . 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 2621 . . . . . 6 ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
3938tgqtop 21425 . . . . 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 692 . . . 4 (𝜑 → ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
41 eqid 2621 . . . . . . 7 (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
4241mopnval 22153 . . . . . 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 6619 . . . 4 (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹))
45 eqid 2621 . . . . . . 7 (MetOpen‘(dist‘𝑈)) = (MetOpen‘(dist‘𝑈))
4645mopnval 22153 . . . . . 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 22044 . . . . . . . 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 6002 . . . . . . 7 ((dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ* → (dist‘𝑈) Fn ((Base‘𝑈) × (Base‘𝑈)))
51 fnresdm 5958 . . . . . . 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 6152 . . . . 5 (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (MetOpen‘(dist‘𝑈)))
543ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝐹:𝑉1-1-onto𝐵)
55 f1of1 6093 . . . . . . . . . . . . . . 15 (𝐹:𝑉1-1-onto𝐵𝐹:𝑉1-1𝐵)
5654, 55syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝐹:𝑉1-1𝐵)
57 cnvimass 5444 . . . . . . . . . . . . . . 15 (𝐹𝑥) ⊆ dom 𝐹
58 f1odm 6098 . . . . . . . . . . . . . . . 16 (𝐹:𝑉1-1-onto𝐵 → dom 𝐹 = 𝑉)
5954, 58syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → dom 𝐹 = 𝑉)
6057, 59syl5sseq 3632 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹𝑥) ⊆ 𝑉)
6114ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
62 simprl 793 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑦𝑉)
63 simprr 795 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑟 ∈ ℝ*)
64 blssm 22133 . . . . . . . . . . . . . . 15 ((((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) ∧ 𝑦𝑉𝑟 ∈ ℝ*) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
6561, 62, 63, 64syl3anc 1323 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
66 f1imaeq 6476 . . . . . . . . . . . . . 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 1321 . . . . . . . . . . . . 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 791 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑥𝐵)
70 foimacnv 6111 . . . . . . . . . . . . . . 15 ((𝐹:𝑉onto𝐵𝑥𝐵) → (𝐹 “ (𝐹𝑥)) = 𝑥)
7168, 69, 70syl2anc 692 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹 “ (𝐹𝑥)) = 𝑥)
721ad2antrr 761 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑈 = (𝐹s 𝑅))
732ad2antrr 761 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑉 = (Base‘𝑅))
744ad2antrr 761 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → 𝑅 ∈ ∞MetSp)
7572, 73, 54, 74, 5, 6, 61, 62, 63imasf1obl 22203 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)))
7675eqcomd 2627 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐵) ∧ (𝑦𝑉𝑟 ∈ ℝ*)) → (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟))
7771, 76eqeq12d 2636 . . . . . . . . . . . . 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 3049 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (∃𝑦𝑉𝑟 ∈ ℝ* (𝐹𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ ∃𝑦𝑉𝑟 ∈ ℝ* 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
803adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → 𝐹:𝑉1-1-onto𝐵)
81 f1ofn 6095 . . . . . . . . . . . 12 (𝐹:𝑉1-1-onto𝐵𝐹 Fn 𝑉)
82 oveq1 6611 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑦) → (𝑧(ball‘(dist‘𝑈))𝑟) = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟))
8382eqeq2d 2631 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑦) → (𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
8483rexbidv 3045 . . . . . . . . . . . . 13 (𝑧 = (𝐹𝑦) → (∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑟 ∈ ℝ* 𝑥 = ((𝐹𝑦)(ball‘(dist‘𝑈))𝑟)))
8584rexrn 6317 . . . . . . . . . . . 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 6075 . . . . . . . . . . . . 13 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
8880, 16, 873syl 18 . . . . . . . . . . . 12 ((𝜑𝑥𝐵) → ran 𝐹 = 𝐵)
8988rexeqdv 3134 . . . . . . . . . . 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 481 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
92 blrn 22124 . . . . . . . . . . 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 481 . . . . . . . . . . 11 ((𝜑𝑥𝐵) → (dist‘𝑈) ∈ (∞Met‘𝐵))
95 blrn 22124 . . . . . . . . . . 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 672 . . . . . . . 8 (𝜑 → ((𝑥𝐵 ∧ (𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) ↔ (𝑥𝐵𝑥 ∈ ran (ball‘(dist‘𝑈)))))
99 f1ofo 6101 . . . . . . . . . 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 21414 . . . . . . . . 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 692 . . . . . . . 8 (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ (𝑥𝐵 ∧ (𝐹𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))))
103 blf 22122 . . . . . . . . . . . 12 ((dist‘𝑈) ∈ (∞Met‘𝐵) → (ball‘(dist‘𝑈)):(𝐵 × ℝ*)⟶𝒫 𝐵)
104 frn 6010 . . . . . . . . . . . 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 3582 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥 ∈ 𝒫 𝐵))
107 elpwi 4140 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
108106, 107syl6 35 . . . . . . . . 9 (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥𝐵))
109108pm4.71rd 666 . . . . . . . 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 2619 . . . . . 6 (𝜑 → (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ran (ball‘(dist‘𝑈)))
112111fveq2d 6152 . . . . 5 (𝜑 → (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)) = (topGen‘ran (ball‘(dist‘𝑈))))
11347, 53, 1123eqtr4d 2665 . . . 4 (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
11440, 44, 1133eqtr4d 2665 . . 3 (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
11526, 31, 1143eqtrd 2659 . 2 (𝜑 → (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
116 eqid 2621 . . 3 (Base‘𝑈) = (Base‘𝑈)
117 eqid 2621 . . 3 ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))
11825, 116, 117isxms2 22163 . 2 (𝑈 ∈ ∞MetSp ↔ (((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)) ∧ (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))))))
11923, 115, 118sylanbrc 697 1 (𝜑𝑈 ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  wss 3555  𝒫 cpw 4130   cuni 4402   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  cres 5076  cima 5077   Fn wfn 5842  wf 5843  1-1wf1 5844  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  *cxr 10017  Basecbs 15781  distcds 15871  TopOpenctopn 16003  topGenctg 16019   qTop cqtop 16084  s cimas 16085  ∞Metcxmt 19650  ballcbl 19652  MetOpencmopn 19655  TopBasesctb 20620  ∞MetSpcxme 22032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-xrs 16083  df-qtop 16088  df-imas 16089  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-bl 19660  df-mopn 19661  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-xms 22035
This theorem is referenced by:  imasf1oms  22205  xpsxms  22249
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