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Theorem nlpfvineqsn 35106
 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 (𝐴𝑉 → ((𝐽 ∈ Top ∧ 𝐴𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑓(𝑓:𝐴𝐽 ∧ ∀𝑝𝐴 ((𝑓𝑝) ∩ 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.
StepHypRef Expression
1 nlpineqsn.x . . . 4 𝑋 = 𝐽
21nlpineqsn 35105 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝𝐴𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝}))
3 simpr 488 . . . . 5 ((𝑝𝑛 ∧ (𝑛𝐴) = {𝑝}) → (𝑛𝐴) = {𝑝})
43reximi 3171 . . . 4 (∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝}) → ∃𝑛𝐽 (𝑛𝐴) = {𝑝})
54ralimi 3092 . . 3 (∀𝑝𝐴𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝}) → ∀𝑝𝐴𝑛𝐽 (𝑛𝐴) = {𝑝})
62, 5syl 17 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝𝐴𝑛𝐽 (𝑛𝐴) = {𝑝})
7 ineq1 4109 . . . 4 (𝑛 = (𝑓𝑝) → (𝑛𝐴) = ((𝑓𝑝) ∩ 𝐴))
87eqeq1d 2760 . . 3 (𝑛 = (𝑓𝑝) → ((𝑛𝐴) = {𝑝} ↔ ((𝑓𝑝) ∩ 𝐴) = {𝑝}))
98ac6sg 9948 . 2 (𝐴𝑉 → (∀𝑝𝐴𝑛𝐽 (𝑛𝐴) = {𝑝} → ∃𝑓(𝑓:𝐴𝐽 ∧ ∀𝑝𝐴 ((𝑓𝑝) ∩ 𝐴) = {𝑝})))
106, 9syl5 34 1 (𝐴𝑉 → ((𝐽 ∈ Top ∧ 𝐴𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑓(𝑓:𝐴𝐽 ∧ ∀𝑝𝐴 ((𝑓𝑝) ∩ 𝐴) = {𝑝})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071   ∩ cin 3857   ⊆ wss 3858  ∅c0 4225  {csn 4522  ∪ cuni 4798  ⟶wf 6331  ‘cfv 6335  Topctop 21593  limPtclp 21834 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-reg 9089  ax-inf2 9137  ax-ac2 9923 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-se 5484  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-isom 6344  df-riota 7108  df-om 7580  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-en 8528  df-r1 9226  df-rank 9227  df-card 9401  df-ac 9576  df-top 21594  df-cld 21719  df-ntr 21720  df-cls 21721  df-lp 21836 This theorem is referenced by: (None)
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