Theorem nlpfvineqsn 38252

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 38251	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
3		simpr 490	. . . . 5 ⊢ ((𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}) → (𝑛 ∩ 𝐴) = {𝑝})
4	3	reximi 3100	. . . 4 ⊢ (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}) → ∃𝑛 ∈ 𝐽 (𝑛 ∩ 𝐴) = {𝑝})
5	4	ralimi 3099	. . 3 ⊢ (∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}) → ∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑛 ∩ 𝐴) = {𝑝})
6	2, 5	syl 18	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑛 ∩ 𝐴) = {𝑝})
7		ineq1 4158	. . . 4 ⊢ (𝑛 = (𝑓‘𝑝) → (𝑛 ∩ 𝐴) = ((𝑓‘𝑝) ∩ 𝐴))
8	7	eqeq1d 2762	. . 3 ⊢ (𝑛 = (𝑓‘𝑝) → ((𝑛 ∩ 𝐴) = {𝑝} ↔ ((𝑓‘𝑝) ∩ 𝐴) = {𝑝}))
9	8	ac6sg 10538	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑛 ∩ 𝐴) = {𝑝} → ∃𝑓(𝑓:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝑓‘𝑝) ∩ 𝐴) = {𝑝})))
10	6, 9	syl5 35	1 ⊢ (𝐴 ∈ 𝑉 → ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑓(𝑓:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝑓‘𝑝) ∩ 𝐴) = {𝑝})))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ∩ cin 3897   ⊆ wss 3898  ∅c0 4278  {csn 4583  ∪ cuni 4866  ⟶wf 6523  ‘cfv 6527  Topctop 23173  limPtclp 23414

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-reg 9564  ax-inf2 9620  ax-ac2 10513

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-en 8952  df-r1 9746  df-rank 9747  df-scott 9901  df-card 9992  df-ac 10167  df-top 23174  df-cld 23299  df-ntr 23300  df-cls 23301  df-lp 23416

This theorem is used by: (None)