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Theorem clsneiel1 43793
Description: If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then membership in the closure of a subset is equivalent to the complement of the subset not being a neighborhood of the point. (Contributed by RP, 7-Jun-2021.)
Hypotheses
Ref Expression
clsnei.o 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
clsnei.p 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
clsnei.d 𝐷 = (𝑃𝐵)
clsnei.f 𝐹 = (𝒫 𝐵𝑂𝐵)
clsnei.h 𝐻 = (𝐹𝐷)
clsnei.r (𝜑𝐾𝐻𝑁)
clsneiel.x (𝜑𝑋𝐵)
clsneiel.s (𝜑𝑆 ∈ 𝒫 𝐵)
Assertion
Ref Expression
clsneiel1 (𝜑 → (𝑋 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚   𝐵,𝑛,𝑜,𝑝   𝐷,𝑖,𝑗,𝑘,𝑙,𝑚   𝐷,𝑛,𝑜,𝑝   𝑖,𝐹,𝑗,𝑘,𝑙   𝑛,𝐹,𝑜,𝑝   𝑖,𝐾,𝑗,𝑘,𝑙,𝑚   𝑛,𝐾,𝑜,𝑝   𝑖,𝑁,𝑗,𝑘,𝑙   𝑛,𝑁,𝑜,𝑝   𝑆,𝑚   𝑆,𝑜   𝑋,𝑙,𝑚   𝜑,𝑖,𝑗,𝑘,𝑙   𝜑,𝑛,𝑜,𝑝
Allowed substitution hints:   𝜑(𝑚)   𝑃(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑆(𝑖,𝑗,𝑘,𝑛,𝑝,𝑙)   𝐹(𝑚)   𝐻(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑁(𝑚)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑋(𝑖,𝑗,𝑘,𝑛,𝑜,𝑝)

Proof of Theorem clsneiel1
StepHypRef Expression
1 clsnei.d . . . 4 𝐷 = (𝑃𝐵)
2 clsnei.h . . . 4 𝐻 = (𝐹𝐷)
3 clsnei.r . . . 4 (𝜑𝐾𝐻𝑁)
41, 2, 3clsneibex 43787 . . 3 (𝜑𝐵 ∈ V)
54ancli 547 . 2 (𝜑 → (𝜑𝐵 ∈ V))
6 clsnei.o . . . . 5 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
7 simpr 483 . . . . . 6 ((𝜑𝐵 ∈ V) → 𝐵 ∈ V)
87pwexd 5385 . . . . 5 ((𝜑𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
9 clsnei.f . . . . 5 𝐹 = (𝒫 𝐵𝑂𝐵)
106, 8, 7, 9fsovfd 43697 . . . 4 ((𝜑𝐵 ∈ V) → 𝐹:(𝒫 𝐵m 𝒫 𝐵)⟶(𝒫 𝒫 𝐵m 𝐵))
1110ffnd 6731 . . 3 ((𝜑𝐵 ∈ V) → 𝐹 Fn (𝒫 𝐵m 𝒫 𝐵))
12 clsnei.p . . . . 5 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
1312, 1, 7dssmapf1od 43706 . . . 4 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
14 f1of 6845 . . . 4 (𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵) → 𝐷:(𝒫 𝐵m 𝒫 𝐵)⟶(𝒫 𝐵m 𝒫 𝐵))
1513, 14syl 17 . . 3 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵m 𝒫 𝐵)⟶(𝒫 𝐵m 𝒫 𝐵))
162breqi 5161 . . . . 5 (𝐾𝐻𝑁𝐾(𝐹𝐷)𝑁)
173, 16sylib 217 . . . 4 (𝜑𝐾(𝐹𝐷)𝑁)
1817adantr 479 . . 3 ((𝜑𝐵 ∈ V) → 𝐾(𝐹𝐷)𝑁)
1911, 15, 18brcoffn 43715 . 2 ((𝜑𝐵 ∈ V) → (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁))
20 simprl 769 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝐾𝐷(𝐷𝐾))
21 clsneiel.x . . . . 5 (𝜑𝑋𝐵)
2221ad2antrr 724 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝑋𝐵)
23 clsneiel.s . . . . 5 (𝜑𝑆 ∈ 𝒫 𝐵)
2423ad2antrr 724 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝑆 ∈ 𝒫 𝐵)
2512, 1, 20, 22, 24ntrclselnel1 43742 . . 3 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝑋 ∈ (𝐾𝑆) ↔ ¬ 𝑋 ∈ ((𝐷𝐾)‘(𝐵𝑆))))
26 simprr 771 . . . . 5 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝐷𝐾)𝐹𝑁)
27 simplr 767 . . . . . 6 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝐵 ∈ V)
28 difssd 4132 . . . . . 6 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝐵𝑆) ⊆ 𝐵)
2927, 28sselpwd 5335 . . . . 5 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝐵𝑆) ∈ 𝒫 𝐵)
306, 9, 26, 22, 29ntrneiel 43766 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝑋 ∈ ((𝐷𝐾)‘(𝐵𝑆)) ↔ (𝐵𝑆) ∈ (𝑁𝑋)))
3130notbid 317 . . 3 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (¬ 𝑋 ∈ ((𝐷𝐾)‘(𝐵𝑆)) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))
3225, 31bitrd 278 . 2 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝑋 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))
335, 19, 32syl2anc2 583 1 (𝜑 → (𝑋 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  {crab 3419  Vcvv 3462  cdif 3944  𝒫 cpw 4607   class class class wbr 5155  cmpt 5238  ccom 5688  wf 6552  1-1-ontowf1o 6555  cfv 6556  (class class class)co 7426  cmpo 7428  m cmap 8857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5292  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-iun 5005  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8005  df-2nd 8006  df-map 8859
This theorem is referenced by:  clsneiel2  43794  clsneifv4  43796
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