MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metcnp3 Structured version   Visualization version   GIF version

Theorem metcnp3 22061
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 21960 . . . 4 (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
323ad2ant1 1074 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4 metcn.4 . . . . 5 𝐾 = (MetOpen‘𝐷)
54mopnval 21959 . . . 4 (𝐷 ∈ (∞Met‘𝑌) → 𝐾 = (topGen‘ran (ball‘𝐷)))
653ad2ant2 1075 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → 𝐾 = (topGen‘ran (ball‘𝐷)))
74mopntopon 21960 . . . 4 (𝐷 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
873ad2ant2 1075 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → 𝐾 ∈ (TopOn‘𝑌))
9 simp3 1055 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → 𝑃𝑋)
103, 6, 8, 9tgcnp 20774 . 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 1093 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝐷 ∈ (∞Met‘𝑌))
12 simplr 787 . . . . . . . . 9 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝐹:𝑋𝑌)
13 simpll3 1094 . . . . . . . . 9 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝑃𝑋)
1412, 13ffvelrnd 6151 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → (𝐹𝑃) ∈ 𝑌)
15 simpr 475 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
16 blcntr 21934 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹𝑃) ∈ 𝑌𝑦 ∈ ℝ+) → (𝐹𝑃) ∈ ((𝐹𝑃)(ball‘𝐷)𝑦))
1711, 14, 15, 16syl3anc 1317 . . . . . . 7 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → (𝐹𝑃) ∈ ((𝐹𝑃)(ball‘𝐷)𝑦))
18 rpxr 11578 . . . . . . . . . 10 (𝑦 ∈ ℝ+𝑦 ∈ ℝ*)
1918adantl 480 . . . . . . . . 9 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ*)
20 blelrn 21938 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹𝑃) ∈ 𝑌𝑦 ∈ ℝ*) → ((𝐹𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷))
2111, 14, 19, 20syl3anc 1317 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → ((𝐹𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷))
22 eleq2 2581 . . . . . . . . . 10 (𝑢 = ((𝐹𝑃)(ball‘𝐷)𝑦) → ((𝐹𝑃) ∈ 𝑢 ↔ (𝐹𝑃) ∈ ((𝐹𝑃)(ball‘𝐷)𝑦)))
23 sseq2 3494 . . . . . . . . . . . 12 (𝑢 = ((𝐹𝑃)(ball‘𝐷)𝑦) → ((𝐹𝑣) ⊆ 𝑢 ↔ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
2423anbi2d 735 . . . . . . . . . . 11 (𝑢 = ((𝐹𝑃)(ball‘𝐷)𝑦) → ((𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢) ↔ (𝑃𝑣 ∧ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))))
2524rexbidv 2938 . . . . . . . . . 10 (𝑢 = ((𝐹𝑃)(ball‘𝐷)𝑦) → (∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢) ↔ ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))))
2622, 25imbi12d 332 . . . . . . . . 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 3182 . . . . . . . 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 42 . . . . . 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 1056 . . . . . . . . . . . 12 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → 𝐶 ∈ (∞Met‘𝑋))
3130ad2antrr 757 . . . . . . . . . . 11 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦 ∈ ℝ+𝑣𝐽)) ∧ 𝑃𝑣) → 𝐶 ∈ (∞Met‘𝑋))
32 simplrr 796 . . . . . . . . . . 11 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦 ∈ ℝ+𝑣𝐽)) ∧ 𝑃𝑣) → 𝑣𝐽)
33 simpr 475 . . . . . . . . . . 11 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦 ∈ ℝ+𝑣𝐽)) ∧ 𝑃𝑣) → 𝑃𝑣)
341mopni2 22014 . . . . . . . . . . 11 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑣𝐽𝑃𝑣) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐶)𝑧) ⊆ 𝑣)
3531, 32, 33, 34syl3anc 1317 . . . . . . . . . 10 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦 ∈ ℝ+𝑣𝐽)) ∧ 𝑃𝑣) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐶)𝑧) ⊆ 𝑣)
36 imass2 5311 . . . . . . . . . . . . 13 ((𝑃(ball‘𝐶)𝑧) ⊆ 𝑣 → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ (𝐹𝑣))
37 sstr2 3479 . . . . . . . . . . . . 13 ((𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ (𝐹𝑣) → ((𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
3836, 37syl 17 . . . . . . . . . . . 12 ((𝑃(ball‘𝐶)𝑧) ⊆ 𝑣 → ((𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
3938com12 32 . . . . . . . . . . 11 ((𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → ((𝑃(ball‘𝐶)𝑧) ⊆ 𝑣 → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
4039reximdv 2903 . . . . . . . . . 10 ((𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) → (∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐶)𝑧) ⊆ 𝑣 → ∃𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
4135, 40syl5com 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)))
4241expimpd 626 . . . . . . . 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)))
4342expr 640 . . . . . . 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))))
4443rexlimdv 2916 . . . . . 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)))
4529, 44syld 45 . . . . 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)))
4645ralrimdva 2856 . . . 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)))
47 simpl2 1057 . . . . . . . . 9 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → 𝐷 ∈ (∞Met‘𝑌))
48 blss 21946 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑌) ∧ 𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹𝑃) ∈ 𝑢) → ∃𝑦 ∈ ℝ+ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢)
49483expib 1259 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑌) → ((𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹𝑃) ∈ 𝑢) → ∃𝑦 ∈ ℝ+ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢))
5047, 49syl 17 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → ((𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹𝑃) ∈ 𝑢) → ∃𝑦 ∈ ℝ+ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢))
51 r19.29r 2959 . . . . . . . . . 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)))
5230ad3antrrr 761 . . . . . . . . . . . . . . . 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))
5313ad2antrr 757 . . . . . . . . . . . . . . . 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)
54 rpxr 11578 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℝ+𝑧 ∈ ℝ*)
5554ad2antrl 759 . . . . . . . . . . . . . . . 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*)
561blopn 22021 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑧 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑧) ∈ 𝐽)
5752, 53, 55, 56syl3anc 1317 . . . . . . . . . . . . . . 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)
58 simprl 789 . . . . . . . . . . . . . . . 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+)
59 blcntr 21934 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑧 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐶)𝑧))
6052, 53, 58, 59syl3anc 1317 . . . . . . . . . . . . . . 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))
61 sstr 3480 . . . . . . . . . . . . . . . . 17 (((𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) ∧ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)
6261ad2ant2l 777 . . . . . . . . . . . . . . . 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)
6362ancoms 467 . . . . . . . . . . . . . . 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)
64 eleq2 2581 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑃(ball‘𝐶)𝑧) → (𝑃𝑣𝑃 ∈ (𝑃(ball‘𝐶)𝑧)))
65 imaeq2 5271 . . . . . . . . . . . . . . . . . 18 (𝑣 = (𝑃(ball‘𝐶)𝑧) → (𝐹𝑣) = (𝐹 “ (𝑃(ball‘𝐶)𝑧)))
6665sseq1d 3499 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑃(ball‘𝐶)𝑧) → ((𝐹𝑣) ⊆ 𝑢 ↔ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢))
6764, 66anbi12d 742 . . . . . . . . . . . . . . . 16 (𝑣 = (𝑃(ball‘𝐶)𝑧) → ((𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢) ↔ (𝑃 ∈ (𝑃(ball‘𝐶)𝑧) ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)))
6867rspcev 3186 . . . . . . . . . . . . . . 15 (((𝑃(ball‘𝐶)𝑧) ∈ 𝐽 ∧ (𝑃 ∈ (𝑃(ball‘𝐶)𝑧) ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)) → ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))
6957, 60, 63, 68syl12anc 1315 . . . . . . . . . . . . . 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))
7069expr 640 . . . . . . . . . . . . 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)))
7170rexlimdva 2917 . . . . . . . . . . . 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)))
7271expimpd 626 . . . . . . . . . . 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)))
7372rexlimdva 2917 . . . . . . . . . 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)))
7451, 73syl5 33 . . . . . . . . 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)))
7574expd 450 . . . . . . . 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))))
7650, 75syld 45 . . . . . . 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))))
7776com23 83 . . . . . 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))))
7877exp4a 630 . . . . 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)))))
7978ralrimdv 2855 . . . 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))))
8046, 79impbid 200 . . 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)))
8180pm5.32da 670 . 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))))
8210, 81bitrd 266 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1938  wral 2800  wrex 2801  wss 3444  ran crn 4933  cima 4935  wf 5685  cfv 5689  (class class class)co 6425  *cxr 9826  +crp 11570  topGenctg 15809  ∞Metcxmt 19460  ballcbl 19462  MetOpencmopn 19465  TopOnctopon 20425   CnP ccnp 20746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721  ax-cnex 9745  ax-resscn 9746  ax-1cn 9747  ax-icn 9748  ax-addcl 9749  ax-addrcl 9750  ax-mulcl 9751  ax-mulrcl 9752  ax-mulcom 9753  ax-addass 9754  ax-mulass 9755  ax-distr 9756  ax-i2m1 9757  ax-1ne0 9758  ax-1rid 9759  ax-rnegex 9760  ax-rrecex 9761  ax-cnre 9762  ax-pre-lttri 9763  ax-pre-lttrn 9764  ax-pre-ltadd 9765  ax-pre-mulgt0 9766  ax-pre-sup 9767
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6387  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-om 6832  df-1st 6932  df-2nd 6933  df-wrecs 7167  df-recs 7229  df-rdg 7267  df-er 7503  df-map 7620  df-en 7716  df-dom 7717  df-sdom 7718  df-sup 8105  df-inf 8106  df-pnf 9829  df-mnf 9830  df-xr 9831  df-ltxr 9832  df-le 9833  df-sub 10017  df-neg 10018  df-div 10432  df-nn 10774  df-2 10832  df-n0 11046  df-z 11117  df-uz 11424  df-q 11527  df-rp 11571  df-xneg 11684  df-xadd 11685  df-xmul 11686  df-topgen 15815  df-psmet 19467  df-xmet 19468  df-bl 19470  df-mopn 19471  df-top 20428  df-bases 20429  df-topon 20430  df-cnp 20749
This theorem is referenced by:  metcnp  22062
  Copyright terms: Public domain W3C validator