Theorem imasf1oxms 23359

Description: The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)

Hypotheses

Ref	Expression
imasf1obl.u	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
imasf1obl.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasf1obl.f	⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
imasf1oxms.r	⊢ (𝜑 → 𝑅 ∈ ∞MetSp)

Assertion

Ref	Expression
imasf1oxms	⊢ (𝜑 → 𝑈 ∈ ∞MetSp)

Proof of Theorem imasf1oxms

Dummy variables 𝑥 𝑟 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasf1obl.u	. . . . 5 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
2		imasf1obl.v	. . . . 5 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasf1obl.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
4		imasf1oxms.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ∞MetSp)
5		eqid 2734	. . . . 5 ⊢ ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
6		eqid 2734	. . . . 5 ⊢ (dist‘𝑈) = (dist‘𝑈)
7		eqid 2734	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2734	. . . . . . . 8 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
9	7, 8	xmsxmet 23326	. . . . . . 7 ⊢ (𝑅 ∈ ∞MetSp → ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) ∈ (∞Met‘(Base‘𝑅)))
10	4, 9	syl 17	. . . . . 6 ⊢ (𝜑 → ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) ∈ (∞Met‘(Base‘𝑅)))
11	2	sqxpeqd 5572	. . . . . . 7 ⊢ (𝜑 → (𝑉 × 𝑉) = ((Base‘𝑅) × (Base‘𝑅)))
12	11	reseq2d 5840	. . . . . 6 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
13	2	fveq2d 6710	. . . . . 6 ⊢ (𝜑 → (∞Met‘𝑉) = (∞Met‘(Base‘𝑅)))
14	10, 12, 13	3eltr4d 2849	. . . . 5 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
15	1, 2, 3, 4, 5, 6, 14	imasf1oxmet 23245	. . . 4 ⊢ (𝜑 → (dist‘𝑈) ∈ (∞Met‘𝐵))
16		f1ofo 6657	. . . . . . 7 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–onto→𝐵)
17	3, 16	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
18	1, 2, 17, 4	imasbas 16989	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
19	18	fveq2d 6710	. . . 4 ⊢ (𝜑 → (∞Met‘𝐵) = (∞Met‘(Base‘𝑈)))
20	15, 19	eleqtrd 2836	. . 3 ⊢ (𝜑 → (dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)))
21		ssid 3913	. . 3 ⊢ (Base‘𝑈) ⊆ (Base‘𝑈)
22		xmetres2 23231	. . 3 ⊢ (((dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)) ∧ (Base‘𝑈) ⊆ (Base‘𝑈)) → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
23	20, 21, 22	sylancl 589	. 2 ⊢ (𝜑 → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
24		eqid 2734	. . . 4 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
25		eqid 2734	. . . 4 ⊢ (TopOpen‘𝑈) = (TopOpen‘𝑈)
26	1, 2, 17, 4, 24, 25	imastopn 22589	. . 3 ⊢ (𝜑 → (TopOpen‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹))
27	24, 7, 8	xmstopn 23321	. . . . . 6 ⊢ (𝑅 ∈ ∞MetSp → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
28	4, 27	syl 17	. . . . 5 ⊢ (𝜑 → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
29	12	fveq2d 6710	. . . . 5 ⊢ (𝜑 → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
30	28, 29	eqtr4d 2777	. . . 4 ⊢ (𝜑 → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))
31	30	oveq1d 7217	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹))
32		blbas 23300	. . . . . 6 ⊢ (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ∈ TopBases)
33	14, 32	syl 17	. . . . 5 ⊢ (𝜑 → ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ∈ TopBases)
34		unirnbl 23290	. . . . . . 7 ⊢ (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = 𝑉)
35		f1oeq2 6639	. . . . . . 7 ⊢ (∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = 𝑉 → (𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto→𝐵 ↔ 𝐹:𝑉–1-1-onto→𝐵))
36	14, 34, 35	3syl 18	. . . . . 6 ⊢ (𝜑 → (𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto→𝐵 ↔ 𝐹:𝑉–1-1-onto→𝐵))
37	3, 36	mpbird 260	. . . . 5 ⊢ (𝜑 → 𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto→𝐵)
38		eqid 2734	. . . . . 6 ⊢ ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
39	38	tgqtop 22581	. . . . 5 ⊢ ((ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ∈ TopBases ∧ 𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto→𝐵) → ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
40	33, 37, 39	syl2anc 587	. . . 4 ⊢ (𝜑 → ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
41		eqid 2734	. . . . . . 7 ⊢ (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
42	41	mopnval 23308	. . . . . 6 ⊢ (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))))
43	14, 42	syl 17	. . . . 5 ⊢ (𝜑 → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))))
44	43	oveq1d 7217	. . . 4 ⊢ (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹))
45		eqid 2734	. . . . . . 7 ⊢ (MetOpen‘(dist‘𝑈)) = (MetOpen‘(dist‘𝑈))
46	45	mopnval 23308	. . . . . 6 ⊢ ((dist‘𝑈) ∈ (∞Met‘𝐵) → (MetOpen‘(dist‘𝑈)) = (topGen‘ran (ball‘(dist‘𝑈))))
47	15, 46	syl 17	. . . . 5 ⊢ (𝜑 → (MetOpen‘(dist‘𝑈)) = (topGen‘ran (ball‘(dist‘𝑈))))
48		xmetf 23199	. . . . . . . 8 ⊢ ((dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)) → (dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ*)
49	20, 48	syl 17	. . . . . . 7 ⊢ (𝜑 → (dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ*)
50		ffn 6534	. . . . . . 7 ⊢ ((dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ* → (dist‘𝑈) Fn ((Base‘𝑈) × (Base‘𝑈)))
51		fnresdm 6485	. . . . . . 7 ⊢ ((dist‘𝑈) Fn ((Base‘𝑈) × (Base‘𝑈)) → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = (dist‘𝑈))
52	49, 50, 51	3syl 18	. . . . . 6 ⊢ (𝜑 → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = (dist‘𝑈))
53	52	fveq2d 6710	. . . . 5 ⊢ (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (MetOpen‘(dist‘𝑈)))
54	3	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝐹:𝑉–1-1-onto→𝐵)
55		f1of1 6649	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–1-1→𝐵)
56	54, 55	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝐹:𝑉–1-1→𝐵)
57		cnvimass 5938	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
58		f1odm 6654	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → dom 𝐹 = 𝑉)
59	54, 58	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → dom 𝐹 = 𝑉)
60	57, 59	sseqtrid 3943	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → (◡𝐹 “ 𝑥) ⊆ 𝑉)
61	14	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
62		simprl 771	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑦 ∈ 𝑉)
63		simprr 773	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑟 ∈ ℝ*)
64		blssm 23288	. . . . . . . . . . . . . . 15 ⊢ ((((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) ∧ 𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
65	61, 62, 63, 64	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
66		f1imaeq 7066	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ ((◡𝐹 “ 𝑥) ⊆ 𝑉 ∧ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)) → ((𝐹 “ (◡𝐹 “ 𝑥)) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) ↔ (◡𝐹 “ 𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)))
67	56, 60, 65, 66	syl12anc 837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → ((𝐹 “ (◡𝐹 “ 𝑥)) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) ↔ (◡𝐹 “ 𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)))
68	54, 16	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝐹:𝑉–onto→𝐵)
69		simplr 769	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑥 ⊆ 𝐵)
70		foimacnv 6667	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝑉–onto→𝐵 ∧ 𝑥 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑥)) = 𝑥)
71	68, 69, 70	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → (𝐹 “ (◡𝐹 “ 𝑥)) = 𝑥)
72	1	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑈 = (𝐹 “s 𝑅))
73	2	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑉 = (Base‘𝑅))
74	4	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑅 ∈ ∞MetSp)
75	72, 73, 54, 74, 5, 6, 61, 62, 63	imasf1obl 23358	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)))
76	75	eqcomd 2740	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟))
77	71, 76	eqeq12d 2750	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → ((𝐹 “ (◡𝐹 “ 𝑥)) = (𝐹 “ (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)) ↔ 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
78	67, 77	bitr3d 284	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → ((◡𝐹 “ 𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
79	78	2rexbidva 3211	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (∃𝑦 ∈ 𝑉 ∃𝑟 ∈ ℝ* (◡𝐹 “ 𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ ∃𝑦 ∈ 𝑉 ∃𝑟 ∈ ℝ* 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
80	3	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝑉–1-1-onto→𝐵)
81		f1ofn 6651	. . . . . . . . . . . 12 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹 Fn 𝑉)
82		oveq1 7209	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑧(ball‘(dist‘𝑈))𝑟) = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟))
83	82	eqeq2d 2745	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
84	83	rexbidv 3209	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑦) → (∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑟 ∈ ℝ* 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
85	84	rexrn 6895	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝑉 → (∃𝑧 ∈ ran 𝐹∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑦 ∈ 𝑉 ∃𝑟 ∈ ℝ* 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
86	80, 81, 85	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (∃𝑧 ∈ ran 𝐹∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑦 ∈ 𝑉 ∃𝑟 ∈ ℝ* 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
87		forn 6625	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
88	80, 16, 87	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → ran 𝐹 = 𝐵)
89	88	rexeqdv 3319	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (∃𝑧 ∈ ran 𝐹∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑧 ∈ 𝐵 ∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
90	79, 86, 89	3bitr2d 310	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (∃𝑦 ∈ 𝑉 ∃𝑟 ∈ ℝ* (◡𝐹 “ 𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ↔ ∃𝑧 ∈ 𝐵 ∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
91	14	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
92		blrn 23279	. . . . . . . . . . 11 ⊢ (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ((◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ↔ ∃𝑦 ∈ 𝑉 ∃𝑟 ∈ ℝ* (◡𝐹 “ 𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)))
93	91, 92	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → ((◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ↔ ∃𝑦 ∈ 𝑉 ∃𝑟 ∈ ℝ* (◡𝐹 “ 𝑥) = (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟)))
94	15	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (dist‘𝑈) ∈ (∞Met‘𝐵))
95		blrn 23279	. . . . . . . . . . 11 ⊢ ((dist‘𝑈) ∈ (∞Met‘𝐵) → (𝑥 ∈ ran (ball‘(dist‘𝑈)) ↔ ∃𝑧 ∈ 𝐵 ∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
96	94, 95	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝑥 ∈ ran (ball‘(dist‘𝑈)) ↔ ∃𝑧 ∈ 𝐵 ∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
97	90, 93, 96	3bitr4d 314	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → ((◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ↔ 𝑥 ∈ ran (ball‘(dist‘𝑈))))
98	97	pm5.32da 582	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ⊆ 𝐵 ∧ (◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) ↔ (𝑥 ⊆ 𝐵 ∧ 𝑥 ∈ ran (ball‘(dist‘𝑈)))))
99		f1ofo 6657	. . . . . . . . . 10 ⊢ (𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto→𝐵 → 𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–onto→𝐵)
100	37, 99	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–onto→𝐵)
101	38	elqtop2 22570	. . . . . . . . 9 ⊢ ((ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ∈ TopBases ∧ 𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–onto→𝐵) → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ (𝑥 ⊆ 𝐵 ∧ (◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))))
102	33, 100, 101	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ (𝑥 ⊆ 𝐵 ∧ (◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))))
103		blf 23277	. . . . . . . . . . . 12 ⊢ ((dist‘𝑈) ∈ (∞Met‘𝐵) → (ball‘(dist‘𝑈)):(𝐵 × ℝ*)⟶𝒫 𝐵)
104		frn 6541	. . . . . . . . . . . 12 ⊢ ((ball‘(dist‘𝑈)):(𝐵 × ℝ*)⟶𝒫 𝐵 → ran (ball‘(dist‘𝑈)) ⊆ 𝒫 𝐵)
105	15, 103, 104	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ran (ball‘(dist‘𝑈)) ⊆ 𝒫 𝐵)
106	105	sseld 3890	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥 ∈ 𝒫 𝐵))
107		elpwi 4512	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝐵 → 𝑥 ⊆ 𝐵)
108	106, 107	syl6 35	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥 ⊆ 𝐵))
109	108	pm4.71rd 566	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) ↔ (𝑥 ⊆ 𝐵 ∧ 𝑥 ∈ ran (ball‘(dist‘𝑈)))))
110	98, 102, 109	3bitr4d 314	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ 𝑥 ∈ ran (ball‘(dist‘𝑈))))
111	110	eqrdv 2732	. . . . . 6 ⊢ (𝜑 → (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ran (ball‘(dist‘𝑈)))
112	111	fveq2d 6710	. . . . 5 ⊢ (𝜑 → (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)) = (topGen‘ran (ball‘(dist‘𝑈))))
113	47, 53, 112	3eqtr4d 2784	. . . 4 ⊢ (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
114	40, 44, 113	3eqtr4d 2784	. . 3 ⊢ (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
115	26, 31, 114	3eqtrd 2778	. 2 ⊢ (𝜑 → (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
116		eqid 2734	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
117		eqid 2734	. . 3 ⊢ ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))
118	25, 116, 117	isxms2 23318	. 2 ⊢ (𝑈 ∈ ∞MetSp ↔ (((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)) ∧ (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))))))
119	23, 115, 118	sylanbrc 586	1 ⊢ (𝜑 → 𝑈 ∈ ∞MetSp)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∃wrex 3055   ⊆ wss 3857  𝒫 cpw 4503  ∪ cuni 4809   × cxp 5538  ^◡ccnv 5539  dom cdm 5540  ran crn 5541   ↾ cres 5542   “ cima 5543   Fn wfn 6364  ⟶wf 6365  –1-1→wf1 6366  –onto→wfo 6367  –1-1-onto→wf1o 6368  ‘cfv 6369  (class class class)co 7202  ℝ^*cxr 10849  Basecbs 16684  distcds 16776  TopOpenctopn 16898  topGenctg 16914   qTop cqtop 16980   “_s cimas 16981  ∞Metcxmet 20320  ballcbl 20322  MetOpencmopn 20325  TopBasesctb 21814  ∞MetSpcxms 23187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789  ax-pre-sup 10790

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-iin 4897  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-se 5499  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-isom 6378  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-of 7458  df-om 7634  df-1st 7750  df-2nd 7751  df-supp 7893  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-er 8380  df-map 8499  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-fsupp 8975  df-sup 9047  df-inf 9048  df-oi 9115  df-card 9538  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-div 11473  df-nn 11814  df-2 11876  df-3 11877  df-4 11878  df-5 11879  df-6 11880  df-7 11881  df-8 11882  df-9 11883  df-n0 12074  df-z 12160  df-dec 12277  df-uz 12422  df-q 12528  df-rp 12570  df-xneg 12687  df-xadd 12688  df-xmul 12689  df-fz 13079  df-fzo 13222  df-seq 13558  df-hash 13880  df-struct 16686  df-ndx 16687  df-slot 16688  df-base 16690  df-sets 16691  df-ress 16692  df-plusg 16780  df-mulr 16781  df-sca 16783  df-vsca 16784  df-ip 16785  df-tset 16786  df-ple 16787  df-ds 16789  df-rest 16899  df-topn 16900  df-0g 16918  df-gsum 16919  df-topgen 16920  df-xrs 16979  df-qtop 16984  df-imas 16985  df-mre 17061  df-mrc 17062  df-acs 17064  df-mgm 18086  df-sgrp 18135  df-mnd 18146  df-submnd 18191  df-mulg 18461  df-cntz 18683  df-cmn 19144  df-psmet 20327  df-xmet 20328  df-bl 20330  df-mopn 20331  df-top 21763  df-topon 21780  df-topsp 21802  df-bases 21815  df-xms 23190

This theorem is referenced by:  imasf1oms  23360  xpsxms  23404