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Theorem metcnp3 23153
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) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(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.
StepHypRef Expression
1 metcn.2 . . . . 5 𝐽 = (MetOpen‘𝐶)
21mopntopon 23052 . . . 4 (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
323ad2ant1 1130 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4 metcn.4 . . . . 5 𝐾 = (MetOpen‘𝐷)
54mopnval 23051 . . . 4 (𝐷 ∈ (∞Met‘𝑌) → 𝐾 = (topGen‘ran (ball‘𝐷)))
653ad2ant2 1131 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → 𝐾 = (topGen‘ran (ball‘𝐷)))
74mopntopon 23052 . . . 4 (𝐷 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
873ad2ant2 1131 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → 𝐾 ∈ (TopOn‘𝑌))
9 simp3 1135 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → 𝑃𝑋)
103, 6, 8, 9tgcnp 21864 . 2 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢 ∈ ran (ball‘𝐷)((𝐹𝑃) ∈ 𝑢 → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))))
11 simpll2 1210 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝐷 ∈ (∞Met‘𝑌))
12 simplr 768 . . . . . . . . 9 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝐹:𝑋𝑌)
13 simpll3 1211 . . . . . . . . 9 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝑃𝑋)
1412, 13ffvelrnd 6843 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → (𝐹𝑃) ∈ 𝑌)
15 simpr 488 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
16 blcntr 23026 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹𝑃) ∈ 𝑌𝑦 ∈ ℝ+) → (𝐹𝑃) ∈ ((𝐹𝑃)(ball‘𝐷)𝑦))
1711, 14, 15, 16syl3anc 1368 . . . . . . 7 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → (𝐹𝑃) ∈ ((𝐹𝑃)(ball‘𝐷)𝑦))
18 rpxr 12395 . . . . . . . . . 10 (𝑦 ∈ ℝ+𝑦 ∈ ℝ*)
1918adantl 485 . . . . . . . . 9 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ*)
20 blelrn 23030 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹𝑃) ∈ 𝑌𝑦 ∈ ℝ*) → ((𝐹𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷))
2111, 14, 19, 20syl3anc 1368 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → ((𝐹𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷))
22 eleq2 2904 . . . . . . . . . 10 (𝑢 = ((𝐹𝑃)(ball‘𝐷)𝑦) → ((𝐹𝑃) ∈ 𝑢 ↔ (𝐹𝑃) ∈ ((𝐹𝑃)(ball‘𝐷)𝑦)))
23 sseq2 3979 . . . . . . . . . . . 12 (𝑢 = ((𝐹𝑃)(ball‘𝐷)𝑦) → ((𝐹𝑣) ⊆ 𝑢 ↔ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
2423anbi2d 631 . . . . . . . . . . 11 (𝑢 = ((𝐹𝑃)(ball‘𝐷)𝑦) → ((𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢) ↔ (𝑃𝑣 ∧ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))))
2524rexbidv 3289 . . . . . . . . . 10 (𝑢 = ((𝐹𝑃)(ball‘𝐷)𝑦) → (∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢) ↔ ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))))
2622, 25imbi12d 348 . . . . . . . . 9 (𝑢 = ((𝐹𝑃)(ball‘𝐷)𝑦) → (((𝐹𝑃) ∈ 𝑢 → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)) ↔ ((𝐹𝑃) ∈ ((𝐹𝑃)(ball‘𝐷)𝑦) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))))
2726rspcv 3604 . . . . . . . 8 (((𝐹𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷) → (∀𝑢 ∈ ran (ball‘𝐷)((𝐹𝑃) ∈ 𝑢 → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)) → ((𝐹𝑃) ∈ ((𝐹𝑃)(ball‘𝐷)𝑦) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))))
2821, 27syl 17 . . . . . . 7 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → (∀𝑢 ∈ ran (ball‘𝐷)((𝐹𝑃) ∈ 𝑢 → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)) → ((𝐹𝑃) ∈ ((𝐹𝑃)(ball‘𝐷)𝑦) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))))
2917, 28mpid 44 . . . . . 6 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → (∀𝑢 ∈ ran (ball‘𝐷)((𝐹𝑃) ∈ 𝑢 → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))))
30 simpl1 1188 . . . . . . . . . . . 12 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → 𝐶 ∈ (∞Met‘𝑋))
3130ad2antrr 725 . . . . . . . . . . 11 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦 ∈ ℝ+𝑣𝐽)) ∧ 𝑃𝑣) → 𝐶 ∈ (∞Met‘𝑋))
32 simplrr 777 . . . . . . . . . . 11 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦 ∈ ℝ+𝑣𝐽)) ∧ 𝑃𝑣) → 𝑣𝐽)
33 simpr 488 . . . . . . . . . . 11 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦 ∈ ℝ+𝑣𝐽)) ∧ 𝑃𝑣) → 𝑃𝑣)
341mopni2 23106 . . . . . . . . . . 11 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑣𝐽𝑃𝑣) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐶)𝑧) ⊆ 𝑣)
3531, 32, 33, 34syl3anc 1368 . . . . . . . . . 10 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦 ∈ ℝ+𝑣𝐽)) ∧ 𝑃𝑣) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐶)𝑧) ⊆ 𝑣)
36 sstr2 3960 . . . . . . . . . . . 12 ((𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ (𝐹𝑣) → ((𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
37 imass2 5952 . . . . . . . . . . . 12 ((𝑃(ball‘𝐶)𝑧) ⊆ 𝑣 → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ (𝐹𝑣))
3836, 37syl11 33 . . . . . . . . . . 11 ((𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → ((𝑃(ball‘𝐶)𝑧) ⊆ 𝑣 → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
3938reximdv 3265 . . . . . . . . . 10 ((𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → (∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐶)𝑧) ⊆ 𝑣 → ∃𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
4035, 39syl5com 31 . . . . . . . . 9 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦 ∈ ℝ+𝑣𝐽)) ∧ 𝑃𝑣) → ((𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → ∃𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
4140expimpd 457 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦 ∈ ℝ+𝑣𝐽)) → ((𝑃𝑣 ∧ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)) → ∃𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
4241expr 460 . . . . . . 7 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → (𝑣𝐽 → ((𝑃𝑣 ∧ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)) → ∃𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))))
4342rexlimdv 3275 . . . . . 6 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → (∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)) → ∃𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
4429, 43syld 47 . . . . 5 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → (∀𝑢 ∈ ran (ball‘𝐷)((𝐹𝑃) ∈ 𝑢 → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)) → ∃𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
4544ralrimdva 3184 . . . 4 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → (∀𝑢 ∈ ran (ball‘𝐷)((𝐹𝑃) ∈ 𝑢 → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)) → ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
46 simpl2 1189 . . . . . . . . 9 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → 𝐷 ∈ (∞Met‘𝑌))
47 blss 23038 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑌) ∧ 𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹𝑃) ∈ 𝑢) → ∃𝑦 ∈ ℝ+ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢)
48473expib 1119 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑌) → ((𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹𝑃) ∈ 𝑢) → ∃𝑦 ∈ ℝ+ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢))
4946, 48syl 17 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → ((𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹𝑃) ∈ 𝑢) → ∃𝑦 ∈ ℝ+ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢))
50 r19.29r 3249 . . . . . . . . . 10 ((∃𝑦 ∈ ℝ+ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)) → ∃𝑦 ∈ ℝ+ (((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢 ∧ ∃𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
5130ad5ant12 755 . . . . . . . . . . . . . . . 16 ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ+ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))) → 𝐶 ∈ (∞Met‘𝑋))
5213ad2antrr 725 . . . . . . . . . . . . . . . 16 ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ+ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))) → 𝑃𝑋)
53 rpxr 12395 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℝ+𝑧 ∈ ℝ*)
5453ad2antrl 727 . . . . . . . . . . . . . . . 16 ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ+ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))) → 𝑧 ∈ ℝ*)
551blopn 23113 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑧 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑧) ∈ 𝐽)
5651, 52, 54, 55syl3anc 1368 . . . . . . . . . . . . . . 15 ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ+ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))) → (𝑃(ball‘𝐶)𝑧) ∈ 𝐽)
57 simprl 770 . . . . . . . . . . . . . . . 16 ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ+ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))) → 𝑧 ∈ ℝ+)
58 blcntr 23026 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑧 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐶)𝑧))
5951, 52, 57, 58syl3anc 1368 . . . . . . . . . . . . . . 15 ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ+ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))) → 𝑃 ∈ (𝑃(ball‘𝐶)𝑧))
60 sstr 3961 . . . . . . . . . . . . . . . . 17 (((𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) ∧ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)
6160ad2ant2l 745 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ ℝ+ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)) ∧ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢)) → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)
6261ancoms 462 . . . . . . . . . . . . . . 15 ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ+ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))) → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)
63 eleq2 2904 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑃(ball‘𝐶)𝑧) → (𝑃𝑣𝑃 ∈ (𝑃(ball‘𝐶)𝑧)))
64 imaeq2 5912 . . . . . . . . . . . . . . . . . 18 (𝑣 = (𝑃(ball‘𝐶)𝑧) → (𝐹𝑣) = (𝐹 “ (𝑃(ball‘𝐶)𝑧)))
6564sseq1d 3984 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑃(ball‘𝐶)𝑧) → ((𝐹𝑣) ⊆ 𝑢 ↔ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢))
6663, 65anbi12d 633 . . . . . . . . . . . . . . . 16 (𝑣 = (𝑃(ball‘𝐶)𝑧) → ((𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢) ↔ (𝑃 ∈ (𝑃(ball‘𝐶)𝑧) ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)))
6766rspcev 3609 . . . . . . . . . . . . . . 15 (((𝑃(ball‘𝐶)𝑧) ∈ 𝐽 ∧ (𝑃 ∈ (𝑃(ball‘𝐶)𝑧) ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))
6856, 59, 62, 67syl12anc 835 . . . . . . . . . . . . . 14 ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ (𝑧 ∈ ℝ+ ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))
6968expr 460 . . . . . . . . . . . . 13 ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) ∧ 𝑧 ∈ ℝ+) → ((𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
7069rexlimdva 3276 . . . . . . . . . . . 12 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) ∧ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) → (∃𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
7170expimpd 457 . . . . . . . . . . 11 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → ((((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢 ∧ ∃𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
7271rexlimdva 3276 . . . . . . . . . 10 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → (∃𝑦 ∈ ℝ+ (((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢 ∧ ∃𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
7350, 72syl5 34 . . . . . . . . 9 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → ((∃𝑦 ∈ ℝ+ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
7473expd 419 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → (∃𝑦 ∈ ℝ+ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢 → (∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))))
7549, 74syld 47 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → ((𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹𝑃) ∈ 𝑢) → (∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))))
7675com23 86 . . . . . 6 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → (∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → ((𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹𝑃) ∈ 𝑢) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))))
7776exp4a 435 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → (∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → (𝑢 ∈ ran (ball‘𝐷) → ((𝐹𝑃) ∈ 𝑢 → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))))
7877ralrimdv 3183 . . . 4 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → (∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → ∀𝑢 ∈ ran (ball‘𝐷)((𝐹𝑃) ∈ 𝑢 → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))))
7945, 78impbid 215 . . 3 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → (∀𝑢 ∈ ran (ball‘𝐷)((𝐹𝑃) ∈ 𝑢 → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)) ↔ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
8079pm5.32da 582 . 2 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → ((𝐹:𝑋𝑌 ∧ ∀𝑢 ∈ ran (ball‘𝐷)((𝐹𝑃) ∈ 𝑢 → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))))
8110, 80bitrd 282 1 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  wrex 3134  wss 3919  ran crn 5543  cima 5545  wf 6339  cfv 6343  (class class class)co 7149  *cxr 10672  +crp 12386  topGenctg 16711  ∞Metcxmet 20132  ballcbl 20134  MetOpencmopn 20137  TopOnctopon 21521   CnP ccnp 21836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-sup 8903  df-inf 8904  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-n0 11895  df-z 11979  df-uz 12241  df-q 12346  df-rp 12387  df-xneg 12504  df-xadd 12505  df-xmul 12506  df-topgen 16717  df-psmet 20139  df-xmet 20140  df-bl 20142  df-mopn 20143  df-top 21505  df-topon 21522  df-bases 21557  df-cnp 21839
This theorem is referenced by:  metcnp  23154
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