Theorem nlpfvineqsn 35580

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 35579	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
3		simpr 485	. . . . 5 ⊢ ((𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}) → (𝑛 ∩ 𝐴) = {𝑝})
4	3	reximi 3178	. . . 4 ⊢ (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}) → ∃𝑛 ∈ 𝐽 (𝑛 ∩ 𝐴) = {𝑝})
5	4	ralimi 3087	. . 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 4139	. . . 4 ⊢ (𝑛 = (𝑓‘𝑝) → (𝑛 ∩ 𝐴) = ((𝑓‘𝑝) ∩ 𝐴))
8	7	eqeq1d 2740	. . 3 ⊢ (𝑛 = (𝑓‘𝑝) → ((𝑛 ∩ 𝐴) = {𝑝} ↔ ((𝑓‘𝑝) ∩ 𝐴) = {𝑝}))
9	8	ac6sg 10244	. 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 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561  ∪ cuni 4839  ⟶wf 6429  ‘cfv 6433  Topctop 22042  limPtclp 22285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399  ax-ac2 10219

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-en 8734  df-r1 9522  df-rank 9523  df-card 9697  df-ac 9872  df-top 22043  df-cld 22170  df-ntr 22171  df-cls 22172  df-lp 22287

This theorem is referenced by: (None)