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Theorem clsneiel1 40809
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 40803 . . 3 (𝜑𝐵 ∈ V)
54ancli 552 . 2 (𝜑 → (𝜑𝐵 ∈ V))
6 clsnei.o . . . . 5 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
7 simpr 488 . . . . . 6 ((𝜑𝐵 ∈ V) → 𝐵 ∈ V)
87pwexd 5245 . . . . 5 ((𝜑𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
9 clsnei.f . . . . 5 𝐹 = (𝒫 𝐵𝑂𝐵)
106, 8, 7, 9fsovfd 40711 . . . 4 ((𝜑𝐵 ∈ V) → 𝐹:(𝒫 𝐵m 𝒫 𝐵)⟶(𝒫 𝒫 𝐵m 𝐵))
1110ffnd 6488 . . 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 40720 . . . 4 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
14 f1of 6590 . . . 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 5036 . . . . 5 (𝐾𝐻𝑁𝐾(𝐹𝐷)𝑁)
173, 16sylib 221 . . . 4 (𝜑𝐾(𝐹𝐷)𝑁)
1817adantr 484 . . 3 ((𝜑𝐵 ∈ V) → 𝐾(𝐹𝐷)𝑁)
1911, 15, 18brcoffn 40731 . 2 ((𝜑𝐵 ∈ V) → (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁))
20 simprl 770 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝐾𝐷(𝐷𝐾))
21 clsneiel.x . . . . 5 (𝜑𝑋𝐵)
2221ad2antrr 725 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝑋𝐵)
23 clsneiel.s . . . . 5 (𝜑𝑆 ∈ 𝒫 𝐵)
2423ad2antrr 725 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝑆 ∈ 𝒫 𝐵)
2512, 1, 20, 22, 24ntrclselnel1 40758 . . 3 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝑋 ∈ (𝐾𝑆) ↔ ¬ 𝑋 ∈ ((𝐷𝐾)‘(𝐵𝑆))))
26 simprr 772 . . . . 5 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝐷𝐾)𝐹𝑁)
27 simplr 768 . . . . . 6 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝐵 ∈ V)
28 difssd 4060 . . . . . 6 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝐵𝑆) ⊆ 𝐵)
2927, 28sselpwd 5194 . . . . 5 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝐵𝑆) ∈ 𝒫 𝐵)
306, 9, 26, 22, 29ntrneiel 40782 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝑋 ∈ ((𝐷𝐾)‘(𝐵𝑆)) ↔ (𝐵𝑆) ∈ (𝑁𝑋)))
3130notbid 321 . . 3 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (¬ 𝑋 ∈ ((𝐷𝐾)‘(𝐵𝑆)) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))
3225, 31bitrd 282 . 2 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝑋 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))
335, 19, 32syl2anc2 588 1 (𝜑 → (𝑋 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  {crab 3110  Vcvv 3441  cdif 3878  𝒫 cpw 4497   class class class wbr 5030  cmpt 5110  ccom 5523  wf 6320  1-1-ontowf1o 6323  cfv 6324  (class class class)co 7135  cmpo 7137  m cmap 8389
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 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391
This theorem is referenced by:  clsneiel2  40810  clsneifv4  40812
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