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Theorem metcnpi 22757

Description: Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 22754. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

**Hypotheses**
Ref	Expression
metcn.2	⊢ 𝐽 = (MetOpen‘𝐶)
metcn.4	⊢ 𝐾 = (MetOpen‘𝐷)

**Assertion**
Ref	Expression
metcnpi	⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) → ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝐴))

Distinct variable groups: 𝑥,𝑦,𝐹 𝑥,𝐽,𝑦 𝑥,𝐾,𝑦 𝑥,𝑋,𝑦 𝑥,𝑌,𝑦 𝑥,𝐴,𝑦 𝑥,𝐶,𝑦 𝑥,𝐷,𝑦 𝑥,𝑃,𝑦

Proof of Theorem metcnpi Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 479	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
2		simpll 757	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐶 ∈ (∞Met‘𝑋))
3		simplr 759	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐷 ∈ (∞Met‘𝑌))
4		eqid 2777	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	cnprcl 21457	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ ∪ 𝐽)
6	5	adantl 475	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ ∪ 𝐽)
7		metcn.2	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘𝐶)
8	7	mopnuni 22654	. . . . . . . 8 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
9	8	ad2antrr 716	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑋 = ∪ 𝐽)
10	6, 9	eleqtrrd 2861	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋)
11		metcn.4	. . . . . . 7 ⊢ 𝐾 = (MetOpen‘𝐷)
12	7, 11	metcnp 22754	. . . . . 6 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑧 ∈ ℝ⁺ ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝑧))))
13	2, 3, 10, 12	syl3anc 1439	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑧 ∈ ℝ⁺ ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝑧))))
14	1, 13	mpbid 224	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑧 ∈ ℝ⁺ ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝑧)))
15	14	simprd 491	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ∀𝑧 ∈ ℝ⁺ ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝑧))
16		breq2 4890	. . . . . 6 ⊢ (𝑧 = 𝐴 → (((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝑧 ↔ ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝐴))
17	16	imbi2d 332	. . . . 5 ⊢ (𝑧 = 𝐴 → (((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝑧) ↔ ((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝐴)))
18	17	rexralbidv 3242	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝑧) ↔ ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝐴)))
19	18	rspccv 3507	. . 3 ⊢ (∀𝑧 ∈ ℝ⁺ ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝑧) → (𝐴 ∈ ℝ⁺ → ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝐴)))
20	15, 19	syl 17	. 2 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐴 ∈ ℝ⁺ → ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝐴)))
21	20	impr 448	1 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) → ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∀wral 3089 ∃wrex 3090 ∪ cuni 4671 class class class wbr 4886 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 < clt 10411 ℝ⁺crp 12137 ∞Metcxmet 20127 MetOpencmopn 20132 CnP ccnp 21437

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-n0 11643 df-z 11729 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-topgen 16490 df-psmet 20134 df-xmet 20135 df-bl 20137 df-mopn 20138 df-top 21106 df-topon 21123 df-bases 21158 df-cnp 21440

This theorem is referenced by: (None)

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