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Theorem metcnp3 23696

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 23592	. . . 4 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
3	2	3ad2ant1 1132	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4		metcn.4	. . . . 5 ⊢ 𝐾 = (MetOpen‘𝐷)
5	4	mopnval 23591	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐾 = (topGen‘ran (ball‘𝐷)))
6	5	3ad2ant2 1133	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝐾 = (topGen‘ran (ball‘𝐷)))
7	4	mopntopon 23592	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
8	7	3ad2ant2 1133	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
9		simp3 1137	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
10	3, 6, 8, 9	tgcnp 22404	. 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 1212	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝐷 ∈ (∞Met‘𝑌))
12		simplr 766	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝐹:𝑋⟶𝑌)
13		simpll3 1213	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝑃 ∈ 𝑋)
14	12, 13	ffvelrnd 6962	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → (𝐹‘𝑃) ∈ 𝑌)
15		simpr 485	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
16		blcntr 23566	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹‘𝑃) ∈ 𝑌 ∧ 𝑦 ∈ ℝ⁺) → (𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦))
17	11, 14, 15, 16	syl3anc 1370	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → (𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦))
18		rpxr 12739	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ^*)
19	18	adantl 482	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ^*)
20		blelrn 23570	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹‘𝑃) ∈ 𝑌 ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷))
21	11, 14, 19, 20	syl3anc 1370	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → ((𝐹‘𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷))
22		eleq2 2827	. . . . . . . . . 10 ⊢ (𝑢 = ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ((𝐹‘𝑃) ∈ 𝑢 ↔ (𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
23		sseq2 3947	. . . . . . . . . . . 12 ⊢ (𝑢 = ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ((𝐹 “ 𝑣) ⊆ 𝑢 ↔ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
24	23	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑢 = ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ((𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢) ↔ (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))
25	24	rexbidv 3226	. . . . . . . . . 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 345	. . . . . . . . 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 3557	. . . . . . . 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 1190	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐶 ∈ (∞Met‘𝑋))
31	30	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) ∧ 𝑃 ∈ 𝑣) → 𝐶 ∈ (∞Met‘𝑋))
32		simplrr 775	. . . . . . . . . . 11 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) ∧ 𝑃 ∈ 𝑣) → 𝑣 ∈ 𝐽)
33		simpr 485	. . . . . . . . . . 11 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) ∧ 𝑃 ∈ 𝑣) → 𝑃 ∈ 𝑣)
34	1	mopni2 23649	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑣 ∈ 𝐽 ∧ 𝑃 ∈ 𝑣) → ∃𝑧 ∈ ℝ⁺ (𝑃(ball‘𝐶)𝑧) ⊆ 𝑣)
35	31, 32, 33, 34	syl3anc 1370	. . . . . . . . . 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 3928	. . . . . . . . . . . 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 6010	. . . . . . . . . . . 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 3202	. . . . . . . . . 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 454	. . . . . . . 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 457	. . . . . . 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 3106	. . . 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 1191	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐷 ∈ (∞Met‘𝑌))
47		blss 23578	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ 𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹‘𝑃) ∈ 𝑢) → ∃𝑦 ∈ ℝ⁺ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢)
48	47	3expib 1121	. . . . . . . . 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 3185	. . . . . . . . . 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	ad5ant12 753	. . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . 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 12739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ^*)
54	53	ad2antrl 725	. . . . . . . . . . . . . . . 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 23656	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ ℝ^*) → (𝑃(ball‘𝐶)𝑧) ∈ 𝐽)
56	51, 52, 54, 55	syl3anc 1370	. . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . 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 23566	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ ℝ⁺) → 𝑃 ∈ (𝑃(ball‘𝐶)𝑧))
59	51, 52, 57, 58	syl3anc 1370	. . . . . . . . . . . . . . 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 3929	. . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑃(ball‘𝐶)𝑧) → (𝑃 ∈ 𝑣 ↔ 𝑃 ∈ (𝑃(ball‘𝐶)𝑧)))
64		imaeq2 5965	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (𝑃(ball‘𝐶)𝑧) → (𝐹 “ 𝑣) = (𝐹 “ (𝑃(ball‘𝐶)𝑧)))
65	64	sseq1d 3952	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑃(ball‘𝐶)𝑧) → ((𝐹 “ 𝑣) ⊆ 𝑢 ↔ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢))
66	63, 65	anbi12d 631	. . . . . . . . . . . . . . . 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 3561	. . . . . . . . . . . . . . 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 834	. . . . . . . . . . . . . 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 457	. . . . . . . . . . . . 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 454	. . . . . . . . . . 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 416	. . . . . . . 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 432	. . . . 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 3105	. . . 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 211	. . 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 579	. 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 278	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 ran crn 5590 “ cima 5592 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℝ^*cxr 11008 ℝ⁺crp 12730 topGenctg 17148 ∞Metcxmet 20582 ballcbl 20584 MetOpencmopn 20587 TopOnctopon 22059 CnP ccnp 22376

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-topgen 17154 df-psmet 20589 df-xmet 20590 df-bl 20592 df-mopn 20593 df-top 22043 df-topon 22060 df-bases 22096 df-cnp 22379

This theorem is referenced by: metcnp 23697

W3C validator