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Theorem imasf1oxms 24835

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⟶1-1onto𝐵)
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⟶1-1onto𝐵)
4		imasf1oxms.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ∞MetSp)
5		eqid 2763	. . . . 5 ⊢ ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
6		eqid 2763	. . . . 5 ⊢ (dist‘𝑈) = (dist‘𝑈)
7		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2763	. . . . . . . 8 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
9	7, 8	xmsxmet 24802	. . . . . . 7 ⊢ (𝑅 ∈ ∞MetSp → ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) ∈ (∞Met‘(Base‘𝑅)))
10	4, 9	syl 18	. . . . . 6 ⊢ (𝜑 → ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) ∈ (∞Met‘(Base‘𝑅)))
11	2	sqxpeqd 5687	. . . . . . 7 ⊢ (𝜑 → (𝑉 × 𝑉) = ((Base‘𝑅) × (Base‘𝑅)))
12	11	reseq2d 5980	. . . . . 6 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
13	2	fveq2d 6909	. . . . . 6 ⊢ (𝜑 → (∞Met‘𝑉) = (∞Met‘(Base‘𝑅)))
14	10, 12, 13	3eltr4d 2879	. . . . 5 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
15	1, 2, 3, 4, 5, 6, 14	imasf1oxmet 24721	. . . 4 ⊢ (𝜑 → (dist‘𝑈) ∈ (∞Met‘𝐵))
16		f1ofo 6851	. . . . . . 7 ⊢ (𝐹:𝑉⟶1-1onto𝐵 → 𝐹:𝑉⟶onto𝐵)
17	3, 16	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑉⟶onto𝐵)
18	1, 2, 17, 4	imasbas 17710	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
19	18	fveq2d 6909	. . . 4 ⊢ (𝜑 → (∞Met‘𝐵) = (∞Met‘(Base‘𝑈)))
20	15, 19	eleqtrd 2866	. . 3 ⊢ (𝜑 → (dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)))
21		ssid 3935	. . 3 ⊢ (Base‘𝑈) ⊆ (Base‘𝑈)
22		xmetres2 24707	. . 3 ⊢ (((dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)) ∧ (Base‘𝑈) ⊆ (Base‘𝑈)) → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
23	20, 21, 22	sylancl 598	. 2 ⊢ (𝜑 → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (∞Met‘(Base‘𝑈)))
24		eqid 2763	. . . 4 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
25		eqid 2763	. . . 4 ⊢ (TopOpen‘𝑈) = (TopOpen‘𝑈)
26	1, 2, 17, 4, 24, 25	imastopn 24066	. . 3 ⊢ (𝜑 → (TopOpen‘𝑈) = ((TopOpen‘𝑅) qTop 𝐹))
27	24, 7, 8	xmstopn 24797	. . . . . 6 ⊢ (𝑅 ∈ ∞MetSp → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
28	4, 27	syl 18	. . . . 5 ⊢ (𝜑 → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
29	12	fveq2d 6909	. . . . 5 ⊢ (𝜑 → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
30	28, 29	eqtr4d 2801	. . . 4 ⊢ (𝜑 → (TopOpen‘𝑅) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))
31	30	oveq1d 7455	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹))
32		blbas 24776	. . . . . 6 ⊢ (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ∈ TopBases)
33	14, 32	syl 18	. . . . 5 ⊢ (𝜑 → ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ∈ TopBases)
34		unirnbl 24766	. . . . . . 7 ⊢ (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = 𝑉)
35		f1oeq2 6832	. . . . . . 7 ⊢ (∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = 𝑉 → (𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))⟶1-1onto𝐵 ↔ 𝐹:𝑉⟶1-1onto𝐵))
36	14, 34, 35	3syl 19	. . . . . 6 ⊢ (𝜑 → (𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))⟶1-1onto𝐵 ↔ 𝐹:𝑉⟶1-1onto𝐵))
37	3, 36	mpbird 260	. . . . 5 ⊢ (𝜑 → 𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))⟶1-1onto𝐵)
38		eqid 2763	. . . . . 6 ⊢ ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = ∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
39	38	tgqtop 24058	. . . . 5 ⊢ ((ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) ∈ TopBases ∧ 𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))⟶1-1onto𝐵) → ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
40	33, 37, 39	syl2anc 596	. . . 4 ⊢ (𝜑 → ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
41		eqid 2763	. . . . . . 7 ⊢ (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))
42	41	mopnval 24784	. . . . . 6 ⊢ (((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉) → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))))
43	14, 42	syl 18	. . . . 5 ⊢ (𝜑 → (MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) = (topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))))
44	43	oveq1d 7455	. . . 4 ⊢ (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ((topGen‘ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) qTop 𝐹))
45		eqid 2763	. . . . . . 7 ⊢ (MetOpen‘(dist‘𝑈)) = (MetOpen‘(dist‘𝑈))
46	45	mopnval 24784	. . . . . 6 ⊢ ((dist‘𝑈) ∈ (∞Met‘𝐵) → (MetOpen‘(dist‘𝑈)) = (topGen‘ran (ball‘(dist‘𝑈))))
47	15, 46	syl 18	. . . . 5 ⊢ (𝜑 → (MetOpen‘(dist‘𝑈)) = (topGen‘ran (ball‘(dist‘𝑈))))
48		xmetf 24675	. . . . . . . 8 ⊢ ((dist‘𝑈) ∈ (∞Met‘(Base‘𝑈)) → (dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ^*)
49	20, 48	syl 18	. . . . . . 7 ⊢ (𝜑 → (dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ^*)
50		ffn 6728	. . . . . . 7 ⊢ ((dist‘𝑈):((Base‘𝑈) × (Base‘𝑈))⟶ℝ^* → (dist‘𝑈) Fn ((Base‘𝑈) × (Base‘𝑈)))
51		fnresdm 6677	. . . . . . 7 ⊢ ((dist‘𝑈) Fn ((Base‘𝑈) × (Base‘𝑈)) → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = (dist‘𝑈))
52	49, 50, 51	3syl 19	. . . . . 6 ⊢ (𝜑 → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = (dist‘𝑈))
53	52	fveq2d 6909	. . . . 5 ⊢ (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (MetOpen‘(dist‘𝑈)))
54	3	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → 𝐹:𝑉⟶1-1onto𝐵)
55		f1of1 6842	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑉⟶1-1onto𝐵 → 𝐹:𝑉⟶1-1𝐵)
56	54, 55	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → 𝐹:𝑉⟶1-1𝐵)
57		cnvimass 6210	. . . . . . . . . . . . . . 15 ⊢ (^◡𝐹 “ 𝑥) ⊆ dom 𝐹
58		f1odm 6847	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑉⟶1-1onto𝐵 → dom 𝐹 = 𝑉)
59	54, 58	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → dom 𝐹 = 𝑉)
60	57, 59	sseqtrid 3955	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → (^◡𝐹 “ 𝑥) ⊆ 𝑉)
61	14	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
62		simprl 783	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → 𝑦 ∈ 𝑉)
63		simprr 785	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^)) → 𝑟 ∈ ℝ^)
64		blssm 24764	. . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → (𝑦(ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))𝑟) ⊆ 𝑉)
66		f1imaeq 7289	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑉⟶1-1𝐵 ∧ ((^◡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 850	. . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → 𝐹:𝑉⟶onto𝐵)
69		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → 𝑥 ⊆ 𝐵)
70		foimacnv 6861	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝑉⟶onto𝐵 ∧ 𝑥 ⊆ 𝐵) → (𝐹 “ (^◡𝐹 “ 𝑥)) = 𝑥)
71	68, 69, 70	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → (𝐹 “ (^◡𝐹 “ 𝑥)) = 𝑥)
72	1	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → 𝑈 = (𝐹 “_s 𝑅))
73	2	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → 𝑉 = (Base‘𝑅))
74	4	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐵) ∧ (𝑦 ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*)) → 𝑅 ∈ ∞MetSp)
75	72, 73, 54, 74, 5, 6, 61, 62, 63	imasf1obl 24834	. . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . 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 3229	. . . . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝑉⟶1-1onto𝐵)
81		f1ofn 6844	. . . . . . . . . . . 12 ⊢ (𝐹:𝑉⟶1-1onto𝐵 → 𝐹 Fn 𝑉)
82		oveq1 7447	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑧(ball‘(dist‘𝑈))𝑟) = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟))
83	82	eqeq2d 2774	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
84	83	rexbidv 3190	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑦) → (∃𝑟 ∈ ℝ^* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟) ↔ ∃𝑟 ∈ ℝ^* 𝑥 = ((𝐹‘𝑦)(ball‘(dist‘𝑈))𝑟)))
85	84	rexrn 7108	. . . . . . . . . . . 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 19	. . . . . . . . . . 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 6818	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑉⟶onto𝐵 → ran 𝐹 = 𝐵)
88	80, 16, 87	3syl 19	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → ran 𝐹 = 𝐵)
89	88	rexeqdv 3324	. . . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (∞Met‘𝑉))
92		blrn 24755	. . . . . . . . . . 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 18	. . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (dist‘𝑈) ∈ (∞Met‘𝐵))
95		blrn 24755	. . . . . . . . . . 11 ⊢ ((dist‘𝑈) ∈ (∞Met‘𝐵) → (𝑥 ∈ ran (ball‘(dist‘𝑈)) ↔ ∃𝑧 ∈ 𝐵 ∃𝑟 ∈ ℝ^* 𝑥 = (𝑧(ball‘(dist‘𝑈))𝑟)))
96	94, 95	syl 18	. . . . . . . . . 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 590	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ⊆ 𝐵 ∧ (^◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))) ↔ (𝑥 ⊆ 𝐵 ∧ 𝑥 ∈ ran (ball‘(dist‘𝑈)))))
99		f1ofo 6851	. . . . . . . . . 10 ⊢ (𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))⟶1-1onto𝐵 → 𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))⟶onto𝐵)
100	37, 99	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:∪ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉)))⟶onto𝐵)
101	38	elqtop2 24047	. . . . . . . . 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) ↔ (𝑥 ⊆ 𝐵 ∧ (^◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))))
102	33, 100, 101	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) ↔ (𝑥 ⊆ 𝐵 ∧ (^◡𝐹 “ 𝑥) ∈ ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))))))
103		blf 24753	. . . . . . . . . . . 12 ⊢ ((dist‘𝑈) ∈ (∞Met‘𝐵) → (ball‘(dist‘𝑈)):(𝐵 × ℝ^*)⟶𝒫 𝐵)
104		frn 6736	. . . . . . . . . . . 12 ⊢ ((ball‘(dist‘𝑈)):(𝐵 × ℝ^*)⟶𝒫 𝐵 → ran (ball‘(dist‘𝑈)) ⊆ 𝒫 𝐵)
105	15, 103, 104	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → ran (ball‘(dist‘𝑈)) ⊆ 𝒫 𝐵)
106	105	sseld 3912	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥 ∈ 𝒫 𝐵))
107		elpwi 4567	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝐵 → 𝑥 ⊆ 𝐵)
108	106, 107	syl6 36	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ran (ball‘(dist‘𝑈)) → 𝑥 ⊆ 𝐵))
109	108	pm4.71rd 572	. . . . . . . 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 2761	. . . . . 6 ⊢ (𝜑 → (ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = ran (ball‘(dist‘𝑈)))
112	111	fveq2d 6909	. . . . 5 ⊢ (𝜑 → (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)) = (topGen‘ran (ball‘(dist‘𝑈))))
113	47, 53, 112	3eqtr4d 2808	. . . 4 ⊢ (𝜑 → (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))) = (topGen‘(ran (ball‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹)))
114	40, 44, 113	3eqtr4d 2808	. . 3 ⊢ (𝜑 → ((MetOpen‘((dist‘𝑅) ↾ (𝑉 × 𝑉))) qTop 𝐹) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
115	26, 31, 114	3eqtrd 2802	. 2 ⊢ (𝜑 → (TopOpen‘𝑈) = (MetOpen‘((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))))
116		eqid 2763	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
117		eqid 2763	. . 3 ⊢ ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈)))
118	25, 116, 117	isxms2 24794	. 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 595	1 ⊢ (𝜑 → 𝑈 ∈ ∞MetSp)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∃wrex 3090 ⊆ wss 3900 𝒫 cpw 4560 ∪ cuni 4870 × cxp 5653 ^◡ccnv 5654 dom cdm 5655 ran crn 5656 ↾ cres 5657 “ cima 5658 Fn wfn 6550 ⟶wf 6551 ⟶1-1wf1 6552 ⟶ontowfo 6553 ⟶1-1ontowf1o 6554 ‘cfv 6555 (class class class)co 7440 ℝ^*cxr 11366 Basecbs 17413 distcds 17463 TopOpenctopn 17618 topGenctg 17634 qTop cqtop 17701 “_s cimas 17702 ∞Metcxmet 21690 ballcbl 21692 MetOpencmopn 21695 TopBasesctb 23290 ∞MetSpcxms 24663

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-pre-sup 11302

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-of 7714 df-om 7899 df-1st 8022 df-2nd 8023 df-supp 8202 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-map 8874 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fsupp 9379 df-sup 9458 df-inf 9459 df-oi 9528 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-uz 12990 df-q 13100 df-rp 13145 df-xneg 13265 df-xadd 13266 df-xmul 13267 df-fz 13664 df-fzo 13813 df-seq 14169 df-hash 14499 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-mulr 17468 df-sca 17470 df-vsca 17471 df-ip 17472 df-tset 17473 df-ple 17474 df-ds 17476 df-rest 17619 df-topn 17620 df-0g 17638 df-gsum 17639 df-topgen 17640 df-xrs 17700 df-qtop 17705 df-imas 17706 df-mre 17782 df-mrc 17783 df-acs 17785 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-submnd 19005 df-mulg 19304 df-cntz 19558 df-cmn 20023 df-psmet 21697 df-xmet 21698 df-bl 21700 df-mopn 21701 df-top 23239 df-topon 23256 df-topsp 23278 df-bases 23291 df-xms 24666

This theorem is used by: imasf1oms 24836 xpsxms 24880

W3C validator