Theorem imasf1oxms 23551

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 2738	. . . . 5 ⊢ ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
6		eqid 2738	. . . . 5 ⊢ (dist‘𝑈) = (dist‘𝑈)
7		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2738	. . . . . . . 8 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
9	7, 8	xmsxmet 23517	. . . . . . 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 5612	. . . . . . 7 ⊢ (𝜑 → (𝑉 × 𝑉) = ((Base‘𝑅) × (Base‘𝑅)))
12	11	reseq2d 5880	. . . . . 6 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
13	2	fveq2d 6760	. . . . . 6 ⊢ (𝜑 → (∞Met‘𝑉) = (∞Met‘(Base‘𝑅)))
14	10, 12, 13	3eltr4d 2854	. . . . 5 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
15	1, 2, 3, 4, 5, 6, 14	imasf1oxmet 23436	. . . 4 ⊢ (𝜑 → (dist‘𝑈) ∈ (∞Met‘𝐵))
16		f1ofo 6707	. . . . . . 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 17140	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
19	18	fveq2d 6760	. . . 4 ⊢ (𝜑 → (∞Met‘𝐵) = (∞Met‘(Base‘𝑈)))
20	15, 19	eleqtrd 2841	. . 3 ⊢ (𝜑 → (dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)))
21		ssid 3939	. . 3 ⊢ (Base‘𝑈) ⊆ (Base‘𝑈)
22		xmetres2 23422	. . 3 ⊢ (((dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)) ∧ (Base‘𝑈) ⊆ (Base‘𝑈)) → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
23	20, 21, 22	sylancl 585	. 2 ⊢ (𝜑 → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
24		eqid 2738	. . . 4 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
25		eqid 2738	. . . 4 ⊢ (TopOpen‘𝑈) = (TopOpen‘𝑈)
26	1, 2, 17, 4, 24, 25	imastopn 22779	. . 3 ⊢ (𝜑 → (TopOpen‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹))
27	24, 7, 8	xmstopn 23512	. . . . . 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 6760	. . . . 5 ⊢ (𝜑 → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
30	28, 29	eqtr4d 2781	. . . 4 ⊢ (𝜑 → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))
31	30	oveq1d 7270	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹))
32		blbas 23491	. . . . . 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 23481	. . . . . . 7 ⊢ (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = 𝑉)
35		f1oeq2 6689	. . . . . . 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 256	. . . . 5 ⊢ (𝜑 → 𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))–1-1-onto→𝐵)
38		eqid 2738	. . . . . 6 ⊢ ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
39	38	tgqtop 22771	. . . . 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 583	. . . 4 ⊢ (𝜑 → ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
41		eqid 2738	. . . . . . 7 ⊢ (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
42	41	mopnval 23499	. . . . . 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 7270	. . . 4 ⊢ (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹))
45		eqid 2738	. . . . . . 7 ⊢ (MetOpen‘(dist‘𝑈)) = (MetOpen‘(dist‘𝑈))
46	45	mopnval 23499	. . . . . 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 23390	. . . . . . . 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 6584	. . . . . . 7 ⊢ ((dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ* → (dist‘𝑈) Fn ((Base‘𝑈) × (Base‘𝑈)))
51		fnresdm 6535	. . . . . . 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 6760	. . . . 5 ⊢ (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (MetOpen‘(dist‘𝑈)))
54	3	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝐹:𝑉–1-1-onto→𝐵)
55		f1of1 6699	. . . . . . . . . . . . . . 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 5978	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
58		f1odm 6704	. . . . . . . . . . . . . . . 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 3969	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → (◡𝐹 “ 𝑥) ⊆ 𝑉)
61	14	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
62		simprl 767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑦 ∈ 𝑉)
63		simprr 769	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑟 ∈ ℝ*)
64		blssm 23479	. . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
66		f1imaeq 7119	. . . . . . . . . . . . . 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 833	. . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑥 ⊆ 𝐵)
70		foimacnv 6717	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝑉–onto→𝐵 ∧ 𝑥 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑥)) = 𝑥)
71	68, 69, 70	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → (𝐹 “ (◡𝐹 “ 𝑥)) = 𝑥)
72	1	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑈 = (𝐹 “s 𝑅))
73	2	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑉 = (Base‘𝑅))
74	4	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ*)) → 𝑅 ∈ ∞MetSp)
75	72, 73, 54, 74, 5, 6, 61, 62, 63	imasf1obl 23550	. . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . 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 2754	. . . . . . . . . . . . 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 280	. . . . . . . . . . . 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 3227	. . . . . . . . . . 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 6701	. . . . . . . . . . . 12 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹 Fn 𝑉)
82		oveq1 7262	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑧(ball‘(dist‘𝑈))𝑟) = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟))
83	82	eqeq2d 2749	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
84	83	rexbidv 3225	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑦) → (∃𝑟 ∈ ℝ* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑟 ∈ ℝ* 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
85	84	rexrn 6945	. . . . . . . . . . . 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 6675	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
88	80, 16, 87	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → ran 𝐹 = 𝐵)
89	88	rexeqdv 3340	. . . . . . . . . . 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 306	. . . . . . . . . 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 23470	. . . . . . . . . . 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 23470	. . . . . . . . . . 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 310	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → ((◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ↔ 𝑥 ∈ ran (ball‘(dist‘𝑈))))
98	97	pm5.32da 578	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ⊆ 𝐵 ∧ (◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) ↔ (𝑥 ⊆ 𝐵 ∧ 𝑥 ∈ ran (ball‘(dist‘𝑈)))))
99		f1ofo 6707	. . . . . . . . . 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 22760	. . . . . . . . 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 583	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ (𝑥 ⊆ 𝐵 ∧ (◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))))
103		blf 23468	. . . . . . . . . . . 12 ⊢ ((dist‘𝑈) ∈ (∞Met‘𝐵) → (ball‘(dist‘𝑈)):(𝐵 × ℝ*)⟶𝒫 𝐵)
104		frn 6591	. . . . . . . . . . . 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 3916	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥 ∈ 𝒫 𝐵))
107		elpwi 4539	. . . . . . . . . 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 310	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ 𝑥 ∈ ran (ball‘(dist‘𝑈))))
111	110	eqrdv 2736	. . . . . 6 ⊢ (𝜑 → (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ran (ball‘(dist‘𝑈)))
112	111	fveq2d 6760	. . . . 5 ⊢ (𝜑 → (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)) = (topGen‘ran (ball‘(dist‘𝑈))))
113	47, 53, 112	3eqtr4d 2788	. . . 4 ⊢ (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
114	40, 44, 113	3eqtr4d 2788	. . 3 ⊢ (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
115	26, 31, 114	3eqtrd 2782	. 2 ⊢ (𝜑 → (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
116		eqid 2738	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
117		eqid 2738	. . 3 ⊢ ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))
118	25, 116, 117	isxms2 23509	. 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 582	1 ⊢ (𝜑 → 𝑈 ∈ ∞MetSp)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836   × cxp 5578  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   ↾ cres 5582   “ cima 5583   Fn wfn 6413  ⟶wf 6414  –1-1→wf1 6415  –onto→wfo 6416  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  ℝ^*cxr 10939  Basecbs 16840  distcds 16897  TopOpenctopn 17049  topGenctg 17065   qTop cqtop 17131   “_s cimas 17132  ∞Metcxmet 20495  ballcbl 20497  MetOpencmopn 20500  TopBasesctb 22003  ∞MetSpcxms 23378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-rest 17050  df-topn 17051  df-0g 17069  df-gsum 17070  df-topgen 17071  df-xrs 17130  df-qtop 17135  df-imas 17136  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-mulg 18616  df-cntz 18838  df-cmn 19303  df-psmet 20502  df-xmet 20503  df-bl 20505  df-mopn 20506  df-top 21951  df-topon 21968  df-topsp 21990  df-bases 22004  df-xms 23381

This theorem is referenced by:  imasf1oms  23552  xpsxms  23596