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Theorem metcnp3 22757

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 22656	. . . 4 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
3	2	3ad2ant1 1124	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4		metcn.4	. . . . 5 ⊢ 𝐾 = (MetOpen‘𝐷)
5	4	mopnval 22655	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐾 = (topGen‘ran (ball‘𝐷)))
6	5	3ad2ant2 1125	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝐾 = (topGen‘ran (ball‘𝐷)))
7	4	mopntopon 22656	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
8	7	3ad2ant2 1125	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
9		simp3 1129	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
10	3, 6, 8, 9	tgcnp 21469	. 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 1228	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝐷 ∈ (∞Met‘𝑌))
12		simplr 759	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝐹:𝑋⟶𝑌)
13		simpll3 1230	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝑃 ∈ 𝑋)
14	12, 13	ffvelrnd 6626	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → (𝐹‘𝑃) ∈ 𝑌)
15		simpr 479	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
16		blcntr 22630	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹‘𝑃) ∈ 𝑌 ∧ 𝑦 ∈ ℝ⁺) → (𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦))
17	11, 14, 15, 16	syl3anc 1439	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → (𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦))
18		rpxr 12152	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ^*)
19	18	adantl 475	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ^*)
20		blelrn 22634	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹‘𝑃) ∈ 𝑌 ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷))
21	11, 14, 19, 20	syl3anc 1439	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → ((𝐹‘𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷))
22		eleq2 2848	. . . . . . . . . 10 ⊢ (𝑢 = ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ((𝐹‘𝑃) ∈ 𝑢 ↔ (𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
23		sseq2 3846	. . . . . . . . . . . 12 ⊢ (𝑢 = ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ((𝐹 “ 𝑣) ⊆ 𝑢 ↔ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
24	23	anbi2d 622	. . . . . . . . . . 11 ⊢ (𝑢 = ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ((𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢) ↔ (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))
25	24	rexbidv 3237	. . . . . . . . . 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 336	. . . . . . . . 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 3507	. . . . . . . 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 17	. . . . . . 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 44	. . . . . 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 1199	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐶 ∈ (∞Met‘𝑋))
31	30	ad2antrr 716	. . . . . . . . . . 11 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) ∧ 𝑃 ∈ 𝑣) → 𝐶 ∈ (∞Met‘𝑋))
32		simplrr 768	. . . . . . . . . . 11 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) ∧ 𝑃 ∈ 𝑣) → 𝑣 ∈ 𝐽)
33		simpr 479	. . . . . . . . . . 11 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) ∧ 𝑃 ∈ 𝑣) → 𝑃 ∈ 𝑣)
34	1	mopni2 22710	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑣 ∈ 𝐽 ∧ 𝑃 ∈ 𝑣) → ∃𝑧 ∈ ℝ⁺ (𝑃(ball‘𝐶)𝑧) ⊆ 𝑣)
35	31, 32, 33, 34	syl3anc 1439	. . . . . . . . . 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 3828	. . . . . . . . . . . 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 5757	. . . . . . . . . . . 12 ⊢ ((𝑃(ball‘𝐶)𝑧) ⊆ 𝑣 → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ (𝐹 “ 𝑣))
38	36, 37	syl11 33	. . . . . . . . . . 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 3197	. . . . . . . . . 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 31	. . . . . . . . 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 447	. . . . . . . 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 450	. . . . . . 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 3212	. . . . . 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 47	. . . . 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 3151	. . . 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 1201	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐷 ∈ (∞Met‘𝑌))
47		blss 22642	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ 𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹‘𝑃) ∈ 𝑢) → ∃𝑦 ∈ ℝ⁺ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢)
48	47	3expib 1113	. . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑌) → ((𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹‘𝑃) ∈ 𝑢) → ∃𝑦 ∈ ℝ⁺ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢))
49	46, 48	syl 17	. . . . . . . 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 3259	. . . . . . . . . 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 720	. . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . 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 12152	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ^*)
54	53	ad2antrl 718	. . . . . . . . . . . . . . . 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 22717	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ ℝ^*) → (𝑃(ball‘𝐶)𝑧) ∈ 𝐽)
56	51, 52, 54, 55	syl3anc 1439	. . . . . . . . . . . . . . 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 761	. . . . . . . . . . . . . . . 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 22630	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ ℝ⁺) → 𝑃 ∈ (𝑃(ball‘𝐶)𝑧))
59	51, 52, 57, 58	syl3anc 1439	. . . . . . . . . . . . . . 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 3829	. . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑃(ball‘𝐶)𝑧) → (𝑃 ∈ 𝑣 ↔ 𝑃 ∈ (𝑃(ball‘𝐶)𝑧)))
64		imaeq2 5718	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (𝑃(ball‘𝐶)𝑧) → (𝐹 “ 𝑣) = (𝐹 “ (𝑃(ball‘𝐶)𝑧)))
65	64	sseq1d 3851	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑃(ball‘𝐶)𝑧) → ((𝐹 “ 𝑣) ⊆ 𝑢 ↔ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢))
66	63, 65	anbi12d 624	. . . . . . . . . . . . . . . 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 3511	. . . . . . . . . . . . . . 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 827	. . . . . . . . . . . . . 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 450	. . . . . . . . . . . . 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 3213	. . . . . . . . . . . 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 447	. . . . . . . . . . 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 3213	. . . . . . . . . 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 34	. . . . . . . . 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 406	. . . . . . . 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 47	. . . . . . 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 86	. . . . . 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 424	. . . . 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 3150	. . . 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 204	. . 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 574	. 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 271	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 setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ∃wrex 3091 ⊆ wss 3792 ran crn 5358 “ cima 5360 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ℝ^*cxr 10412 ℝ⁺crp 12141 topGenctg 16488 ∞Metcxmet 20131 ballcbl 20133 MetOpencmopn 20136 TopOnctopon 21126 CnP ccnp 21441

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-sup 8638 df-inf 8639 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-n0 11647 df-z 11733 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-topgen 16494 df-psmet 20138 df-xmet 20139 df-bl 20141 df-mopn 20142 df-top 21110 df-topon 21127 df-bases 21162 df-cnp 21444

This theorem is referenced by: metcnp 22758

W3C validator