Theorem nlpfvineqsn 37784

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 37783	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
3		simpr 486	. . . . 5 ⊢ ((𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}) → (𝑛 ∩ 𝐴) = {𝑝})
4	3	reximi 3079	. . . 4 ⊢ (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}) → ∃𝑛 ∈ 𝐽 (𝑛 ∩ 𝐴) = {𝑝})
5	4	ralimi 3078	. . 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 4144	. . . 4 ⊢ (𝑛 = (𝑓‘𝑝) → (𝑛 ∩ 𝐴) = ((𝑓‘𝑝) ∩ 𝐴))
8	7	eqeq1d 2743	. . 3 ⊢ (𝑛 = (𝑓‘𝑝) → ((𝑛 ∩ 𝐴) = {𝑝} ↔ ((𝑓‘𝑝) ∩ 𝐴) = {𝑝}))
9	8	ac6sg 10406	. 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 397   ∧ w3a 1093   = wceq 1548  ∃wex 1787   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065   ∩ cin 3883   ⊆ wss 3884  ∅c0 4263  {csn 4557  ∪ cuni 4840  ⟶wf 6484  ‘cfv 6488  Topctop 22879  limPtclp 23120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681  ax-reg 9501  ax-inf2 9557  ax-ac2 10381

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-iin 4926  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-isom 6497  df-riota 7316  df-ov 7362  df-om 7810  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-en 8888  df-r1 9683  df-rank 9684  df-card 9858  df-ac 10033  df-top 22880  df-cld 23005  df-ntr 23006  df-cls 23007  df-lp 23122

This theorem is referenced by: (None)