Theorem imasf1oxms 24377

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 2729	. . . . 5 ⊢ ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
6		eqid 2729	. . . . 5 ⊢ (dist‘𝑈) = (dist‘𝑈)
7		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2729	. . . . . . . 8 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
9	7, 8	xmsxmet 24344	. . . . . . 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 5670	. . . . . . 7 ⊢ (𝜑 → (𝑉 × 𝑉) = ((Base‘𝑅) × (Base‘𝑅)))
12	11	reseq2d 5950	. . . . . 6 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
13	2	fveq2d 6862	. . . . . 6 ⊢ (𝜑 → (∞Met‘𝑉) = (∞Met‘(Base‘𝑅)))
14	10, 12, 13	3eltr4d 2843	. . . . 5 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
15	1, 2, 3, 4, 5, 6, 14	imasf1oxmet 24263	. . . 4 ⊢ (𝜑 → (dist‘𝑈) ∈ (∞Met‘𝐵))
16		f1ofo 6807	. . . . . . 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 17475	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
19	18	fveq2d 6862	. . . 4 ⊢ (𝜑 → (∞Met‘𝐵) = (∞Met‘(Base‘𝑈)))
20	15, 19	eleqtrd 2830	. . 3 ⊢ (𝜑 → (dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)))
21		ssid 3969	. . 3 ⊢ (Base‘𝑈) ⊆ (Base‘𝑈)
22		xmetres2 24249	. . 3 ⊢ (((dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)) ∧ (Base‘𝑈) ⊆ (Base‘𝑈)) → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
23	20, 21, 22	sylancl 586	. 2 ⊢ (𝜑 → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
24		eqid 2729	. . . 4 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
25		eqid 2729	. . . 4 ⊢ (TopOpen‘𝑈) = (TopOpen‘𝑈)
26	1, 2, 17, 4, 24, 25	imastopn 23607	. . 3 ⊢ (𝜑 → (TopOpen‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹))
27	24, 7, 8	xmstopn 24339	. . . . . 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 6862	. . . . 5 ⊢ (𝜑 → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
30	28, 29	eqtr4d 2767	. . . 4 ⊢ (𝜑 → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))
31	30	oveq1d 7402	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹))
32		blbas 24318	. . . . . 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 24308	. . . . . . 7 ⊢ (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = 𝑉)
35		f1oeq2 6789	. . . . . . 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 257	. . . . 5 ⊢ (𝜑 → 𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto→𝐵)
38		eqid 2729	. . . . . 6 ⊢ ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
39	38	tgqtop 23599	. . . . 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 584	. . . 4 ⊢ (𝜑 → ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
41		eqid 2729	. . . . . . 7 ⊢ (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
42	41	mopnval 24326	. . . . . 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 7402	. . . 4 ⊢ (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹))
45		eqid 2729	. . . . . . 7 ⊢ (MetOpen‘(dist‘𝑈)) = (MetOpen‘(dist‘𝑈))
46	45	mopnval 24326	. . . . . 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 24217	. . . . . . . 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 6688	. . . . . . 7 ⊢ ((dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ* → (dist‘𝑈) Fn ((Base‘𝑈) × (Base‘𝑈)))
51		fnresdm 6637	. . . . . . 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 6862	. . . . 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 6799	. . . . . . . . . . . . . . 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 6053	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
58		f1odm 6804	. . . . . . . . . . . . . . . 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 3989	. . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑦 ∈ 𝑉)
63		simprr 772	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑟 ∈ ℝ*)
64		blssm 24306	. . . . . . . . . . . . . . 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 7240	. . . . . . . . . . . . . 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 836	. . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑥 ⊆ 𝐵)
70		foimacnv 6817	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝑉–onto→𝐵 ∧ 𝑥 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑥)) = 𝑥)
71	68, 69, 70	syl2anc 584	. . . . . . . . . . . . . 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 24376	. . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . 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 281	. . . . . . . . . . . 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 3200	. . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝑉–1-1-onto→𝐵)
81		f1ofn 6801	. . . . . . . . . . . 12 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹 Fn 𝑉)
82		oveq1 7394	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑧(ball‘(dist‘𝑈))𝑟) = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟))
83	82	eqeq2d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
84	83	rexbidv 3157	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑦) → (∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑟 ∈ ℝ* 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
85	84	rexrn 7059	. . . . . . . . . . . 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 6775	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
88	80, 16, 87	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → ran 𝐹 = 𝐵)
89	88	rexeqdv 3300	. . . . . . . . . . 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 307	. . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
92		blrn 24297	. . . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (dist‘𝑈) ∈ (∞Met‘𝐵))
95		blrn 24297	. . . . . . . . . . 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 311	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → ((◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ↔ 𝑥 ∈ ran (ball‘(dist‘𝑈))))
98	97	pm5.32da 579	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ⊆ 𝐵 ∧ (◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) ↔ (𝑥 ⊆ 𝐵 ∧ 𝑥 ∈ ran (ball‘(dist‘𝑈)))))
99		f1ofo 6807	. . . . . . . . . 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 23588	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ (𝑥 ⊆ 𝐵 ∧ (◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))))
103		blf 24295	. . . . . . . . . . . 12 ⊢ ((dist‘𝑈) ∈ (∞Met‘𝐵) → (ball‘(dist‘𝑈)):(𝐵 × ℝ*)⟶𝒫 𝐵)
104		frn 6695	. . . . . . . . . . . 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 3945	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥 ∈ 𝒫 𝐵))
107		elpwi 4570	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝐵 → 𝑥 ⊆ 𝐵)
108	106, 107	syl6 35	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥 ⊆ 𝐵))
109	108	pm4.71rd 562	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) ↔ (𝑥 ⊆ 𝐵 ∧ 𝑥 ∈ ran (ball‘(dist‘𝑈)))))
110	98, 102, 109	3bitr4d 311	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ 𝑥 ∈ ran (ball‘(dist‘𝑈))))
111	110	eqrdv 2727	. . . . . 6 ⊢ (𝜑 → (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ran (ball‘(dist‘𝑈)))
112	111	fveq2d 6862	. . . . 5 ⊢ (𝜑 → (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)) = (topGen‘ran (ball‘(dist‘𝑈))))
113	47, 53, 112	3eqtr4d 2774	. . . 4 ⊢ (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
114	40, 44, 113	3eqtr4d 2774	. . 3 ⊢ (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
115	26, 31, 114	3eqtrd 2768	. 2 ⊢ (𝜑 → (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
116		eqid 2729	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
117		eqid 2729	. . 3 ⊢ ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))
118	25, 116, 117	isxms2 24336	. 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 583	1 ⊢ (𝜑 → 𝑈 ∈ ∞MetSp)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871   × cxp 5636  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641   Fn wfn 6506  ⟶wf 6507  –1-1→wf1 6508  –onto→wfo 6509  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387  ℝ^*cxr 11207  Basecbs 17179  distcds 17229  TopOpenctopn 17384  topGenctg 17400   qTop cqtop 17466   “_s cimas 17467  ∞Metcxmet 21249  ballcbl 21251  MetOpencmopn 21254  TopBasesctb 22832  ∞MetSpcxms 24205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-sup 9393  df-inf 9394  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-q 12908  df-rp 12952  df-xneg 13072  df-xadd 13073  df-xmul 13074  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-rest 17385  df-topn 17386  df-0g 17404  df-gsum 17405  df-topgen 17406  df-xrs 17465  df-qtop 17470  df-imas 17471  df-mre 17547  df-mrc 17548  df-acs 17550  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-mulg 19000  df-cntz 19249  df-cmn 19712  df-psmet 21256  df-xmet 21257  df-bl 21259  df-mopn 21260  df-top 22781  df-topon 22798  df-topsp 22820  df-bases 22833  df-xms 24208

This theorem is referenced by:  imasf1oms  24378  xpsxms  24422