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Theorem metcnp3 14307

Description: Two ways to express that 𝐹 is continuous at 𝑃 for metric spaces. Proposition 14-4.2 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

**Hypotheses**
Ref	Expression
metcn.2	⊢ 𝐽 = (MetOpen‘𝐶)
metcn.4	⊢ 𝐾 = (MetOpen‘𝐷)

**Assertion**
Ref	Expression
metcnp3	⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))

Distinct variable groups: 𝑦,𝑧,𝐹 𝑦,𝐽,𝑧 𝑦,𝐾,𝑧 𝑦,𝑋,𝑧 𝑦,𝑌,𝑧 𝑦,𝐶,𝑧 𝑦,𝐷,𝑧 𝑦,𝑃,𝑧

Proof of Theorem metcnp3 Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metcn.2	. . . . 5 ⊢ 𝐽 = (MetOpen‘𝐶)
2	1	mopntopon 14239	. . . 4 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
3	2	3ad2ant1 1019	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4		metcn.4	. . . . 5 ⊢ 𝐾 = (MetOpen‘𝐷)
5	4	mopnval 14238	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐾 = (topGen‘ran (ball‘𝐷)))
6	5	3ad2ant2 1020	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝐾 = (topGen‘ran (ball‘𝐷)))
7	4	mopntopon 14239	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
8	7	3ad2ant2 1020	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
9		simp3 1000	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
10	3, 6, 8, 9	tgcnp 14005	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ ran (ball‘𝐷)((𝐹‘𝑃) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))))
11		simpll2 1038	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝐷 ∈ (∞Met‘𝑌))
12		simplr 528	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝐹:𝑋⟶𝑌)
13		simpll3 1039	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝑃 ∈ 𝑋)
14	12, 13	ffvelcdmd 5665	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → (𝐹‘𝑃) ∈ 𝑌)
15		simpr 110	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
16		blcntr 14212	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹‘𝑃) ∈ 𝑌 ∧ 𝑦 ∈ ℝ⁺) → (𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦))
17	11, 14, 15, 16	syl3anc 1248	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → (𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦))
18		rpxr 9675	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ^*)
19	18	adantl 277	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ^*)
20		blelrn 14216	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹‘𝑃) ∈ 𝑌 ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷))
21	11, 14, 19, 20	syl3anc 1248	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → ((𝐹‘𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷))
22		eleq2 2251	. . . . . . . . . 10 ⊢ (𝑢 = ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ((𝐹‘𝑃) ∈ 𝑢 ↔ (𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
23		sseq2 3191	. . . . . . . . . . . 12 ⊢ (𝑢 = ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ((𝐹 “ 𝑣) ⊆ 𝑢 ↔ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
24	23	anbi2d 464	. . . . . . . . . . 11 ⊢ (𝑢 = ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ((𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢) ↔ (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))
25	24	rexbidv 2488	. . . . . . . . . 10 ⊢ (𝑢 = ((𝐹‘𝑃)(ball‘𝐷)𝑦) → (∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢) ↔ ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))
26	22, 25	imbi12d 234	. . . . . . . . 9 ⊢ (𝑢 = ((𝐹‘𝑃)(ball‘𝐷)𝑦) → (((𝐹‘𝑃) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)) ↔ ((𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))))
27	26	rspcv 2849	. . . . . . . 8 ⊢ (((𝐹‘𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷) → (∀𝑢 ∈ ran (ball‘𝐷)((𝐹‘𝑃) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)) → ((𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))))
28	21, 27	syl 14	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → (∀𝑢 ∈ ran (ball‘𝐷)((𝐹‘𝑃) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)) → ((𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))))
29	17, 28	mpid 42	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → (∀𝑢 ∈ ran (ball‘𝐷)((𝐹‘𝑃) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))
30		simpl1 1001	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐶 ∈ (∞Met‘𝑋))
31	30	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) ∧ 𝑃 ∈ 𝑣) → 𝐶 ∈ (∞Met‘𝑋))
32		simplrr 536	. . . . . . . . . . 11 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) ∧ 𝑃 ∈ 𝑣) → 𝑣 ∈ 𝐽)
33		simpr 110	. . . . . . . . . . 11 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) ∧ 𝑃 ∈ 𝑣) → 𝑃 ∈ 𝑣)
34	1	mopni2 14279	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑣 ∈ 𝐽 ∧ 𝑃 ∈ 𝑣) → ∃𝑧 ∈ ℝ⁺ (𝑃(ball‘𝐶)𝑧) ⊆ 𝑣)
35	31, 32, 33, 34	syl3anc 1248	. . . . . . . . . 10 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) ∧ 𝑃 ∈ 𝑣) → ∃𝑧 ∈ ℝ⁺ (𝑃(ball‘𝐶)𝑧) ⊆ 𝑣)
36		sstr2 3174	. . . . . . . . . . . 12 ⊢ ((𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ (𝐹 “ 𝑣) → ((𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
37		imass2 5016	. . . . . . . . . . . 12 ⊢ ((𝑃(ball‘𝐶)𝑧) ⊆ 𝑣 → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ (𝐹 “ 𝑣))
38	36, 37	syl11 31	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ((𝑃(ball‘𝐶)𝑧) ⊆ 𝑣 → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
39	38	reximdv 2588	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) → (∃𝑧 ∈ ℝ⁺ (𝑃(ball‘𝐶)𝑧) ⊆ 𝑣 → ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
40	35, 39	syl5com 29	. . . . . . . . 9 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) ∧ 𝑃 ∈ 𝑣) → ((𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
41	40	expimpd 363	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) → ((𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)) → ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
42	41	expr 375	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → (𝑣 ∈ 𝐽 → ((𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)) → ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))
43	42	rexlimdv 2603	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → (∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)) → ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
44	29, 43	syld 45	. . . . 5 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → (∀𝑢 ∈ ran (ball‘𝐷)((𝐹‘𝑃) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)) → ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
45	44	ralrimdva 2567	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑢 ∈ ran (ball‘𝐷)((𝐹‘𝑃) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)) → ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
46		simpl2 1002	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐷 ∈ (∞Met‘𝑌))
47		blss 14224	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ 𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹‘𝑃) ∈ 𝑢) → ∃𝑦 ∈ ℝ⁺ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢)
48	47	3expib 1207	. . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑌) → ((𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹‘𝑃) ∈ 𝑢) → ∃𝑦 ∈ ℝ⁺ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢))
49	46, 48	syl 14	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹‘𝑃) ∈ 𝑢) → ∃𝑦 ∈ ℝ⁺ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢))
50		r19.29r 2625	. . . . . . . . . 10 ⊢ ((∃𝑦 ∈ ℝ⁺ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢 ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)) → ∃𝑦 ∈ ℝ⁺ (((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢 ∧ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
51	30	ad3antrrr 492	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ⁺ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))) → 𝐶 ∈ (∞Met‘𝑋))
52	13	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ⁺ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))) → 𝑃 ∈ 𝑋)
53		rpxr 9675	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ^*)
54	53	ad2antrl 490	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ⁺ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))) → 𝑧 ∈ ℝ^*)
55	1	blopn 14286	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ ℝ^*) → (𝑃(ball‘𝐶)𝑧) ∈ 𝐽)
56	51, 52, 54, 55	syl3anc 1248	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ⁺ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))) → (𝑃(ball‘𝐶)𝑧) ∈ 𝐽)
57		simprl 529	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ⁺ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))) → 𝑧 ∈ ℝ⁺)
58		blcntr 14212	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ ℝ⁺) → 𝑃 ∈ (𝑃(ball‘𝐶)𝑧))
59	51, 52, 57, 58	syl3anc 1248	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ⁺ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))) → 𝑃 ∈ (𝑃(ball‘𝐶)𝑧))
60		sstr 3175	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ∧ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)
61	60	ad2ant2l 508	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ ℝ⁺ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)) ∧ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢)) → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)
62	61	ancoms 268	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ⁺ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))) → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)
63		eleq2 2251	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑃(ball‘𝐶)𝑧) → (𝑃 ∈ 𝑣 ↔ 𝑃 ∈ (𝑃(ball‘𝐶)𝑧)))
64		imaeq2 4978	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (𝑃(ball‘𝐶)𝑧) → (𝐹 “ 𝑣) = (𝐹 “ (𝑃(ball‘𝐶)𝑧)))
65	64	sseq1d 3196	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑃(ball‘𝐶)𝑧) → ((𝐹 “ 𝑣) ⊆ 𝑢 ↔ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢))
66	63, 65	anbi12d 473	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = (𝑃(ball‘𝐶)𝑧) → ((𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢) ↔ (𝑃 ∈ (𝑃(ball‘𝐶)𝑧) ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)))
67	66	rspcev 2853	. . . . . . . . . . . . . . 15 ⊢ (((𝑃(ball‘𝐶)𝑧) ∈ 𝐽 ∧ (𝑃 ∈ (𝑃(ball‘𝐶)𝑧) ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢))
68	56, 59, 62, 67	syl12anc 1246	. . . . . . . . . . . . . 14 ⊢ ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ⁺ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢))
69	68	expr 375	. . . . . . . . . . . . 13 ⊢ ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ 𝑧 ∈ ℝ⁺) → ((𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
70	69	rexlimdva 2604	. . . . . . . . . . . 12 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) → (∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
71	70	expimpd 363	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → ((((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢 ∧ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
72	71	rexlimdva 2604	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∃𝑦 ∈ ℝ⁺ (((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢 ∧ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
73	50, 72	syl5 32	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((∃𝑦 ∈ ℝ⁺ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢 ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
74	73	expd 258	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∃𝑦 ∈ ℝ⁺ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢 → (∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢))))
75	49, 74	syld 45	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹‘𝑃) ∈ 𝑢) → (∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢))))
76	75	com23 78	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ((𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹‘𝑃) ∈ 𝑢) → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢))))
77	76	exp4a 366	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) → (𝑢 ∈ ran (ball‘𝐷) → ((𝐹‘𝑃) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))))
78	77	ralrimdv 2566	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ∀𝑢 ∈ ran (ball‘𝐷)((𝐹‘𝑃) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢))))
79	45, 78	impbid 129	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑢 ∈ ran (ball‘𝐷)((𝐹‘𝑃) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)) ↔ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
80	79	pm5.32da 452	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ ran (ball‘𝐷)((𝐹‘𝑃) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢))) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))
81	10, 80	bitrd 188	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 979 = wceq 1363 ∈ wcel 2158 ∀wral 2465 ∃wrex 2466 ⊆ wss 3141 ran crn 4639 “ cima 4641 ⟶wf 5224 ‘cfv 5228 (class class class)co 5888 ℝ^*cxr 8005 ℝ⁺crp 9667 topGenctg 12721 ∞Metcxmet 13722 ballcbl 13724 MetOpencmopn 13727 TopOnctopon 13806 CnP ccnp 13982

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-mulrcl 7924 ax-addcom 7925 ax-mulcom 7926 ax-addass 7927 ax-mulass 7928 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-1rid 7932 ax-0id 7933 ax-rnegex 7934 ax-precex 7935 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-apti 7940 ax-pre-ltadd 7941 ax-pre-mulgt0 7942 ax-pre-mulext 7943 ax-arch 7944 ax-caucvg 7945

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-recs 6320 df-frec 6406 df-map 6664 df-sup 6997 df-inf 6998 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-sub 8144 df-neg 8145 df-reap 8546 df-ap 8553 df-div 8644 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 df-n0 9191 df-z 9268 df-uz 9543 df-q 9634 df-rp 9668 df-xneg 9786 df-xadd 9787 df-seqfrec 10460 df-exp 10534 df-cj 10865 df-re 10866 df-im 10867 df-rsqrt 11021 df-abs 11022 df-topgen 12727 df-psmet 13729 df-xmet 13730 df-bl 13732 df-mopn 13733 df-top 13794 df-topon 13807 df-bases 13839 df-cnp 13985

This theorem is referenced by: metcnp 14308

W3C validator