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Theorem metcnp3 22155
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 22054 . . . 4 (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
323ad2ant1 1075 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4 metcn.4 . . . . 5 𝐾 = (MetOpen‘𝐷)
54mopnval 22053 . . . 4 (𝐷 ∈ (∞Met‘𝑌) → 𝐾 = (topGen‘ran (ball‘𝐷)))
653ad2ant2 1076 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → 𝐾 = (topGen‘ran (ball‘𝐷)))
74mopntopon 22054 . . . 4 (𝐷 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
873ad2ant2 1076 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → 𝐾 ∈ (TopOn‘𝑌))
9 simp3 1056 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → 𝑃𝑋)
103, 6, 8, 9tgcnp 20867 . 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 1094 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝐷 ∈ (∞Met‘𝑌))
12 simplr 788 . . . . . . . . 9 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝐹:𝑋𝑌)
13 simpll3 1095 . . . . . . . . 9 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝑃𝑋)
1412, 13ffvelrnd 6268 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → (𝐹𝑃) ∈ 𝑌)
15 simpr 476 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
16 blcntr 22028 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹𝑃) ∈ 𝑌𝑦 ∈ ℝ+) → (𝐹𝑃) ∈ ((𝐹𝑃)(ball‘𝐷)𝑦))
1711, 14, 15, 16syl3anc 1318 . . . . . . 7 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → (𝐹𝑃) ∈ ((𝐹𝑃)(ball‘𝐷)𝑦))
18 rpxr 11716 . . . . . . . . . 10 (𝑦 ∈ ℝ+𝑦 ∈ ℝ*)
1918adantl 481 . . . . . . . . 9 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ*)
20 blelrn 22032 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹𝑃) ∈ 𝑌𝑦 ∈ ℝ*) → ((𝐹𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷))
2111, 14, 19, 20syl3anc 1318 . . . . . . . 8 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ 𝑦 ∈ ℝ+) → ((𝐹𝑃)(ball‘𝐷)𝑦) ∈ ran (ball‘𝐷))
22 eleq2 2677 . . . . . . . . . 10 (𝑢 = ((𝐹𝑃)(ball‘𝐷)𝑦) → ((𝐹𝑃) ∈ 𝑢 ↔ (𝐹𝑃) ∈ ((𝐹𝑃)(ball‘𝐷)𝑦)))
23 sseq2 3590 . . . . . . . . . . . 12 (𝑢 = ((𝐹𝑃)(ball‘𝐷)𝑦) → ((𝐹𝑣) ⊆ 𝑢 ↔ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦)))
2423anbi2d 736 . . . . . . . . . . 11 (𝑢 = ((𝐹𝑃)(ball‘𝐷)𝑦) → ((𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢) ↔ (𝑃𝑣 ∧ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))))
2524rexbidv 3034 . . . . . . . . . 10 (𝑢 = ((𝐹𝑃)(ball‘𝐷)𝑦) → (∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢) ↔ ∃𝑣𝐽 (𝑃𝑣 ∧ (𝐹𝑣) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))))
2622, 25imbi12d 333 . . . . . . . . 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 3278 . . . . . . . 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 43 . . . . . 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 1057 . . . . . . . . . . . 12 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → 𝐶 ∈ (∞Met‘𝑋))
3130ad2antrr 758 . . . . . . . . . . 11 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦 ∈ ℝ+𝑣𝐽)) ∧ 𝑃𝑣) → 𝐶 ∈ (∞Met‘𝑋))
32 simplrr 797 . . . . . . . . . . 11 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦 ∈ ℝ+𝑣𝐽)) ∧ 𝑃𝑣) → 𝑣𝐽)
33 simpr 476 . . . . . . . . . . 11 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦 ∈ ℝ+𝑣𝐽)) ∧ 𝑃𝑣) → 𝑃𝑣)
341mopni2 22108 . . . . . . . . . . 11 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑣𝐽𝑃𝑣) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐶)𝑧) ⊆ 𝑣)
3531, 32, 33, 34syl3anc 1318 . . . . . . . . . 10 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) ∧ (𝑦 ∈ ℝ+𝑣𝐽)) ∧ 𝑃𝑣) → ∃𝑧 ∈ ℝ+ (𝑃(ball‘𝐶)𝑧) ⊆ 𝑣)
36 imass2 5420 . . . . . . . . . . . . 13 ((𝑃(ball‘𝐶)𝑧) ⊆ 𝑣 → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ (𝐹𝑣))
37 sstr2 3575 . . . . . . . . . . . . 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 2999 . . . . . . . . . 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 627 . . . . . . . 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 641 . . . . . . 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 3012 . . . . . 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 46 . . . . 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 2952 . . . 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 1058 . . . . . . . . 9 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) ∧ 𝐹:𝑋𝑌) → 𝐷 ∈ (∞Met‘𝑌))
48 blss 22040 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑌) ∧ 𝑢 ∈ ran (ball‘𝐷) ∧ (𝐹𝑃) ∈ 𝑢) → ∃𝑦 ∈ ℝ+ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢)
49483expib 1260 . . . . . . . . 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 3055 . . . . . . . . . 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 762 . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . 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 11716 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℝ+𝑧 ∈ ℝ*)
5554ad2antrl 760 . . . . . . . . . . . . . . . 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 22115 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑧 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑧) ∈ 𝐽)
5752, 53, 55, 56syl3anc 1318 . . . . . . . . . . . . . . 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 790 . . . . . . . . . . . . . . . 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 22028 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑧 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐶)𝑧))
6052, 53, 58, 59syl3anc 1318 . . . . . . . . . . . . . . 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 3576 . . . . . . . . . . . . . . . . 17 (((𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦) ∧ ((𝐹𝑃)(ball‘𝐷)𝑦) ⊆ 𝑢) → (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)
6261ad2ant2l 778 . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . 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 2677 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑃(ball‘𝐶)𝑧) → (𝑃𝑣𝑃 ∈ (𝑃(ball‘𝐶)𝑧)))
65 imaeq2 5381 . . . . . . . . . . . . . . . . . 18 (𝑣 = (𝑃(ball‘𝐶)𝑧) → (𝐹𝑣) = (𝐹 “ (𝑃(ball‘𝐶)𝑧)))
6665sseq1d 3595 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑃(ball‘𝐶)𝑧) → ((𝐹𝑣) ⊆ 𝑢 ↔ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢))
6764, 66anbi12d 743 . . . . . . . . . . . . . . . 16 (𝑣 = (𝑃(ball‘𝐶)𝑧) → ((𝑃𝑣 ∧ (𝐹𝑣) ⊆ 𝑢) ↔ (𝑃 ∈ (𝑃(ball‘𝐶)𝑧) ∧ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ 𝑢)))
6867rspcev 3282 . . . . . . . . . . . . . . 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 1316 . . . . . . . . . . . . . 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 641 . . . . . . . . . . . . 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 3013 . . . . . . . . . . . 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 627 . . . . . . . . . . 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 3013 . . . . . . . . . 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 451 . . . . . . . 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 46 . . . . . . 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 84 . . . . . 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 631 . . . . 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 2951 . . . 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 201 . . 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 671 . 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 267 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897  wss 3540  ran crn 5039  cima 5041  wf 5800  cfv 5804  (class class class)co 6549  *cxr 9952  +crp 11708  topGenctg 15921  ∞Metcxmt 19552  ballcbl 19554  MetOpencmopn 19557  TopOnctopon 20518   CnP ccnp 20839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cnp 20842
This theorem is referenced by:  metcnp  22156
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