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Theorem metcnp3 24498

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 24395	. . . 4 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
3	2	3ad2ant1 1133	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4		metcn.4	. . . . 5 ⊢ 𝐾 = (MetOpen‘𝐷)
5	4	mopnval 24394	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐾 = (topGen‘ran (ball‘𝐷)))
6	5	3ad2ant2 1134	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝐾 = (topGen‘ran (ball‘𝐷)))
7	4	mopntopon 24395	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
8	7	3ad2ant2 1134	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
9		simp3 1138	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
10	3, 6, 8, 9	tgcnp 23208	. 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 1213	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝐷 ∈ (∞Met‘𝑌))
12		simplr 768	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝐹:𝑋⟶𝑌)
13		simpll3 1214	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝑃 ∈ 𝑋)
14	12, 13	ffvelcdmd 7085	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → (𝐹‘𝑃) ∈ 𝑌)
15		simpr 484	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
16		blcntr 24369	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹‘𝑃) ∈ 𝑌 ∧ 𝑦 ∈ ℝ⁺) → (𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦))
17	11, 14, 15, 16	syl3anc 1372	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → (𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦))
18		rpxr 13026	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ^*)
19	18	adantl 481	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ^*)
20		blelrn 24373	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹‘𝑃) ∈ 𝑌 ∧ 𝑦 ∈ ℝ^*) → ((𝐹‘𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷))
21	11, 14, 19, 20	syl3anc 1372	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → ((𝐹‘𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷))
22		eleq2 2822	. . . . . . . . . 10 ⊢ (𝑢 = ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ((𝐹‘𝑃) ∈ 𝑢 ↔ (𝐹‘𝑃) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
23		sseq2 3990	. . . . . . . . . . . 12 ⊢ (𝑢 = ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ((𝐹 “ 𝑣) ⊆ 𝑢 ↔ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
24	23	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑢 = ((𝐹‘𝑃)(ball‘𝐷)𝑦) → ((𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢) ↔ (𝑃 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))
25	24	rexbidv 3166	. . . . . . . . . 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 344	. . . . . . . . 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 3601	. . . . . . . 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 1191	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐶 ∈ (∞Met‘𝑋))
31	30	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) ∧ 𝑃 ∈ 𝑣) → 𝐶 ∈ (∞Met‘𝑋))
32		simplrr 777	. . . . . . . . . . 11 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) ∧ 𝑃 ∈ 𝑣) → 𝑣 ∈ 𝐽)
33		simpr 484	. . . . . . . . . . 11 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑣 ∈ 𝐽)) ∧ 𝑃 ∈ 𝑣) → 𝑃 ∈ 𝑣)
34	1	mopni2 24451	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑣 ∈ 𝐽 ∧ 𝑃 ∈ 𝑣) → ∃𝑧 ∈ ℝ⁺ (𝑃(ball‘𝐶)𝑧) ⊆ 𝑣)
35	31, 32, 33, 34	syl3anc 1372	. . . . . . . . . 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 3970	. . . . . . . . . . . 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 6100	. . . . . . . . . . . 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 3157	. . . . . . . . . 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 453	. . . . . . . 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 456	. . . . . . 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 3140	. . . . . 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 3141	. . . 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 1192	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐷 ∈ (∞Met‘𝑌))
47		blss 24381	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ 𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹‘𝑃) ∈ 𝑢) → ∃𝑦 ∈ ℝ⁺ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢)
48	47	3expib 1122	. . . . . . . . 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 3103	. . . . . . . . . 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 730	. . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . 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 13026	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ^*)
54	53	ad2antrl 728	. . . . . . . . . . . . . . . 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 24458	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ ℝ^*) → (𝑃(ball‘𝐶)𝑧) ∈ 𝐽)
56	51, 52, 54, 55	syl3anc 1372	. . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . 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 24369	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ ℝ⁺) → 𝑃 ∈ (𝑃(ball‘𝐶)𝑧))
59	51, 52, 57, 58	syl3anc 1372	. . . . . . . . . . . . . . 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 3972	. . . . . . . . . . . . . . . . 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 746	. . . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑃(ball‘𝐶)𝑧) → (𝑃 ∈ 𝑣 ↔ 𝑃 ∈ (𝑃(ball‘𝐶)𝑧)))
64		imaeq2 6054	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (𝑃(ball‘𝐶)𝑧) → (𝐹 “ 𝑣) = (𝐹 “ (𝑃(ball‘𝐶)𝑧)))
65	64	sseq1d 3995	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑃(ball‘𝐶)𝑧) → ((𝐹 “ 𝑣) ⊆ 𝑢 ↔ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢))
66	63, 65	anbi12d 632	. . . . . . . . . . . . . . . 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 3605	. . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . 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 456	. . . . . . . . . . . . 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 3142	. . . . . . . . . . . 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 453	. . . . . . . . . . 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 3142	. . . . . . . . . 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 415	. . . . . . . 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 431	. . . . 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 3139	. . . 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 212	. . 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 279	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3050 ∃wrex 3059 ⊆ wss 3931 ran crn 5666 “ cima 5668 ⟶wf 6537 ‘cfv 6541 (class class class)co 7413 ℝ^*cxr 11276 ℝ⁺crp 13016 topGenctg 17454 ∞Metcxmet 21312 ballcbl 21314 MetOpencmopn 21317 TopOnctopon 22865 CnP ccnp 23180

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214 ax-pre-sup 11215

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-1st 7996 df-2nd 7997 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8727 df-map 8850 df-en 8968 df-dom 8969 df-sdom 8970 df-sup 9464 df-inf 9465 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-div 11903 df-nn 12249 df-2 12311 df-n0 12510 df-z 12597 df-uz 12861 df-q 12973 df-rp 13017 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-topgen 17460 df-psmet 21319 df-xmet 21320 df-bl 21322 df-mopn 21323 df-top 22849 df-topon 22866 df-bases 22901 df-cnp 23183

This theorem is referenced by: metcnp 24499

W3C validator