Theorem nlpfvineqsn 37553

Description: Given a subset 𝐴 of 𝑋 with no limit points, there exists a function from each point 𝑝 of 𝐴 to an open set intersecting 𝐴 only at 𝑝. This proof uses the axiom of choice. (Contributed by ML, 23-Mar-2021.)

Hypothesis

Ref	Expression
nlpineqsn.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
nlpfvineqsn	⊢ (𝐴 ∈ 𝑉 → ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑓(𝑓:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝑓‘𝑝) ∩ 𝐴) = {𝑝})))

Distinct variable groups:   𝐴,𝑝   𝐽,𝑝   𝑋,𝑝   𝐴,𝑓,𝑝   𝑓,𝐽

Allowed substitution hints:   𝑉(𝑓,𝑝)   𝑋(𝑓)

Proof of Theorem nlpfvineqsn

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nlpineqsn.x	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	nlpineqsn 37552	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
3		simpr 484	. . . . 5 ⊢ ((𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}) → (𝑛 ∩ 𝐴) = {𝑝})
4	3	reximi 3072	. . . 4 ⊢ (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}) → ∃𝑛 ∈ 𝐽 (𝑛 ∩ 𝐴) = {𝑝})
5	4	ralimi 3071	. . 3 ⊢ (∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}) → ∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑛 ∩ 𝐴) = {𝑝})
6	2, 5	syl 17	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑛 ∩ 𝐴) = {𝑝})
7		ineq1 4163	. . . 4 ⊢ (𝑛 = (𝑓‘𝑝) → (𝑛 ∩ 𝐴) = ((𝑓‘𝑝) ∩ 𝐴))
8	7	eqeq1d 2736	. . 3 ⊢ (𝑛 = (𝑓‘𝑝) → ((𝑛 ∩ 𝐴) = {𝑝} ↔ ((𝑓‘𝑝) ∩ 𝐴) = {𝑝}))
9	8	ac6sg 10396	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑛 ∩ 𝐴) = {𝑝} → ∃𝑓(𝑓:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝑓‘𝑝) ∩ 𝐴) = {𝑝})))
10	6, 9	syl5 34	1 ⊢ (𝐴 ∈ 𝑉 → ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑓(𝑓:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝑓‘𝑝) ∩ 𝐴) = {𝑝})))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058   ∩ cin 3898   ⊆ wss 3899  ∅c0 4283  {csn 4578  ∪ cuni 4861  ⟶wf 6486  ‘cfv 6490  Topctop 22835  limPtclp 23076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-reg 9495  ax-inf2 9548  ax-ac2 10371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-en 8882  df-r1 9674  df-rank 9675  df-card 9849  df-ac 10024  df-top 22836  df-cld 22961  df-ntr 22962  df-cls 22963  df-lp 23078

This theorem is referenced by: (None)