Theorem nlpfvineqsn 35507

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 35506	. . 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 3174	. . . 4 ⊢ (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}) → ∃𝑛 ∈ 𝐽 (𝑛 ∩ 𝐴) = {𝑝})
5	4	ralimi 3086	. . 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 4136	. . . 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 10175	. 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 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558  ∪ cuni 4836  ⟶wf 6414  ‘cfv 6418  Topctop 21950  limPtclp 22193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-inf2 9329  ax-ac2 10150

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-en 8692  df-r1 9453  df-rank 9454  df-card 9628  df-ac 9803  df-top 21951  df-cld 22078  df-ntr 22079  df-cls 22080  df-lp 22195

This theorem is referenced by: (None)