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Theorem imasf1oxms 24005

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 2732	. . . . 5 ⊢ ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
6		eqid 2732	. . . . 5 ⊢ (dist‘𝑈) = (dist‘𝑈)
7		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2732	. . . . . . . 8 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
9	7, 8	xmsxmet 23969	. . . . . . 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 5708	. . . . . . 7 ⊢ (𝜑 → (𝑉 × 𝑉) = ((Base‘𝑅) × (Base‘𝑅)))
12	11	reseq2d 5981	. . . . . 6 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
13	2	fveq2d 6895	. . . . . 6 ⊢ (𝜑 → (∞Met‘𝑉) = (∞Met‘(Base‘𝑅)))
14	10, 12, 13	3eltr4d 2848	. . . . 5 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
15	1, 2, 3, 4, 5, 6, 14	imasf1oxmet 23888	. . . 4 ⊢ (𝜑 → (dist‘𝑈) ∈ (∞Met‘𝐵))
16		f1ofo 6840	. . . . . . 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 17460	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
19	18	fveq2d 6895	. . . 4 ⊢ (𝜑 → (∞Met‘𝐵) = (∞Met‘(Base‘𝑈)))
20	15, 19	eleqtrd 2835	. . 3 ⊢ (𝜑 → (dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)))
21		ssid 4004	. . 3 ⊢ (Base‘𝑈) ⊆ (Base‘𝑈)
22		xmetres2 23874	. . 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 2732	. . . 4 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
25		eqid 2732	. . . 4 ⊢ (TopOpen‘𝑈) = (TopOpen‘𝑈)
26	1, 2, 17, 4, 24, 25	imastopn 23231	. . 3 ⊢ (𝜑 → (TopOpen‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹))
27	24, 7, 8	xmstopn 23964	. . . . . 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 6895	. . . . 5 ⊢ (𝜑 → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
30	28, 29	eqtr4d 2775	. . . 4 ⊢ (𝜑 → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))
31	30	oveq1d 7426	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹))
32		blbas 23943	. . . . . 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 23933	. . . . . . 7 ⊢ (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = 𝑉)
35		f1oeq2 6822	. . . . . . 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 2732	. . . . . 6 ⊢ ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
39	38	tgqtop 23223	. . . . 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 2732	. . . . . . 7 ⊢ (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
42	41	mopnval 23951	. . . . . 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 7426	. . . 4 ⊢ (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹))
45		eqid 2732	. . . . . . 7 ⊢ (MetOpen‘(dist‘𝑈)) = (MetOpen‘(dist‘𝑈))
46	45	mopnval 23951	. . . . . 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 23842	. . . . . . . 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 6717	. . . . . . 7 ⊢ ((dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ^* → (dist‘𝑈) Fn ((Base‘𝑈) × (Base‘𝑈)))
51		fnresdm 6669	. . . . . . 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 6895	. . . . 5 ⊢ (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (MetOpen‘(dist‘𝑈)))
54	3	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → 𝐹:𝑉–1-1-onto→𝐵)
55		f1of1 6832	. . . . . . . . . . . . . . 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 6080	. . . . . . . . . . . . . . 15 ⊢ (^◡𝐹 “ 𝑥) ⊆ dom 𝐹
58		f1odm 6837	. . . . . . . . . . . . . . . 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 4034	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → (^◡𝐹 “ 𝑥) ⊆ 𝑉)
61	14	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
62		simprl 769	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → 𝑦 ∈ 𝑉)
63		simprr 771	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^)) → 𝑟 ∈ ℝ^)
64		blssm 23931	. . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
66		f1imaeq 7266	. . . . . . . . . . . . . 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 835	. . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → 𝑥 ⊆ 𝐵)
70		foimacnv 6850	. . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → 𝑈 = (𝐹 “_s 𝑅))
73	2	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → 𝑉 = (Base‘𝑅))
74	4	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → 𝑅 ∈ ∞MetSp)
75	72, 73, 54, 74, 5, 6, 61, 62, 63	imasf1obl 24004	. . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . 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 3217	. . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝑉–1-1-onto→𝐵)
81		f1ofn 6834	. . . . . . . . . . . 12 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹 Fn 𝑉)
82		oveq1 7418	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑧(ball‘(dist‘𝑈))𝑟) = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟))
83	82	eqeq2d 2743	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
84	83	rexbidv 3178	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑦) → (∃𝑟 ∈ ℝ^* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑟 ∈ ℝ^* 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
85	84	rexrn 7088	. . . . . . . . . . . 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 6808	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
88	80, 16, 87	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → ran 𝐹 = 𝐵)
89	88	rexeqdv 3326	. . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
92		blrn 23922	. . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (dist‘𝑈) ∈ (∞Met‘𝐵))
95		blrn 23922	. . . . . . . . . . 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 579	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ⊆ 𝐵 ∧ (^◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) ↔ (𝑥 ⊆ 𝐵 ∧ 𝑥 ∈ ran (ball‘(dist‘𝑈)))))
99		f1ofo 6840	. . . . . . . . . 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 23212	. . . . . . . . 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 23920	. . . . . . . . . . . 12 ⊢ ((dist‘𝑈) ∈ (∞Met‘𝐵) → (ball‘(dist‘𝑈)):(𝐵 × ℝ^*)⟶𝒫 𝐵)
104		frn 6724	. . . . . . . . . . . 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 3981	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥 ∈ 𝒫 𝐵))
107		elpwi 4609	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝐵 → 𝑥 ⊆ 𝐵)
108	106, 107	syl6 35	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥 ⊆ 𝐵))
109	108	pm4.71rd 563	. . . . . . . 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 2730	. . . . . 6 ⊢ (𝜑 → (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ran (ball‘(dist‘𝑈)))
112	111	fveq2d 6895	. . . . 5 ⊢ (𝜑 → (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)) = (topGen‘ran (ball‘(dist‘𝑈))))
113	47, 53, 112	3eqtr4d 2782	. . . 4 ⊢ (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
114	40, 44, 113	3eqtr4d 2782	. . 3 ⊢ (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
115	26, 31, 114	3eqtrd 2776	. 2 ⊢ (𝜑 → (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
116		eqid 2732	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
117		eqid 2732	. . 3 ⊢ ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))
118	25, 116, 117	isxms2 23961	. 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)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ⊆ wss 3948 𝒫 cpw 4602 ∪ cuni 4908 × cxp 5674 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 ↾ cres 5678 “ cima 5679 Fn wfn 6538 ⟶wf 6539 –1-1→wf1 6540 –onto→wfo 6541 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 ℝ^*cxr 11249 Basecbs 17146 distcds 17208 TopOpenctopn 17369 topGenctg 17385 qTop cqtop 17451 “_s cimas 17452 ∞Metcxmet 20935 ballcbl 20937 MetOpencmopn 20940 TopBasesctb 22455 ∞MetSpcxms 23830

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-fz 13487 df-fzo 13630 df-seq 13969 df-hash 14293 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-xrs 17450 df-qtop 17455 df-imas 17456 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-mulg 18953 df-cntz 19183 df-cmn 19652 df-psmet 20942 df-xmet 20943 df-bl 20945 df-mopn 20946 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-xms 23833

This theorem is referenced by: imasf1oms 24006 xpsxms 24050

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