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Theorem clsneiel1 39821
 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 39815 . . 3 (𝜑𝐵 ∈ V)
54ancli 541 . 2 (𝜑 → (𝜑𝐵 ∈ V))
6 clsnei.o . . . . . 6 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
7 simpr 477 . . . . . . 7 ((𝜑𝐵 ∈ V) → 𝐵 ∈ V)
87pwexd 5134 . . . . . 6 ((𝜑𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
9 clsnei.f . . . . . 6 𝐹 = (𝒫 𝐵𝑂𝐵)
106, 8, 7, 9fsovfd 39721 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)⟶(𝒫 𝒫 𝐵𝑚 𝐵))
1110ffnd 6347 . . . 4 ((𝜑𝐵 ∈ V) → 𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵))
12 clsnei.p . . . . . 6 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
1312, 1, 7dssmapf1od 39730 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
14 f1of 6446 . . . . 5 (𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵) → 𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)⟶(𝒫 𝐵𝑚 𝒫 𝐵))
1513, 14syl 17 . . . 4 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)⟶(𝒫 𝐵𝑚 𝒫 𝐵))
162breqi 4936 . . . . . 6 (𝐾𝐻𝑁𝐾(𝐹𝐷)𝑁)
173, 16sylib 210 . . . . 5 (𝜑𝐾(𝐹𝐷)𝑁)
1817adantr 473 . . . 4 ((𝜑𝐵 ∈ V) → 𝐾(𝐹𝐷)𝑁)
1911, 15, 18brcoffn 39743 . . 3 ((𝜑𝐵 ∈ V) → (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁))
2019ancli 541 . 2 ((𝜑𝐵 ∈ V) → ((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)))
21 simprl 758 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝐾𝐷(𝐷𝐾))
22 clsneiel.x . . . . 5 (𝜑𝑋𝐵)
2322ad2antrr 713 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝑋𝐵)
24 clsneiel.s . . . . 5 (𝜑𝑆 ∈ 𝒫 𝐵)
2524ad2antrr 713 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝑆 ∈ 𝒫 𝐵)
2612, 1, 21, 23, 25ntrclselnel1 39770 . . 3 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝑋 ∈ (𝐾𝑆) ↔ ¬ 𝑋 ∈ ((𝐷𝐾)‘(𝐵𝑆))))
27 simprr 760 . . . . 5 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝐷𝐾)𝐹𝑁)
28 simplr 756 . . . . . 6 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝐵 ∈ V)
29 difssd 4001 . . . . . 6 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝐵𝑆) ⊆ 𝐵)
3028, 29sselpwd 5087 . . . . 5 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝐵𝑆) ∈ 𝒫 𝐵)
316, 9, 27, 23, 30ntrneiel 39794 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝑋 ∈ ((𝐷𝐾)‘(𝐵𝑆)) ↔ (𝐵𝑆) ∈ (𝑁𝑋)))
3231notbid 310 . . 3 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (¬ 𝑋 ∈ ((𝐷𝐾)‘(𝐵𝑆)) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))
3326, 32bitrd 271 . 2 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝑋 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))
345, 20, 333syl 18 1 (𝜑 → (𝑋 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  {crab 3092  Vcvv 3415   ∖ cdif 3828  𝒫 cpw 4423   class class class wbr 4930   ↦ cmpt 5009   ∘ ccom 5412  ⟶wf 6186  –1-1-onto→wf1o 6189  ‘cfv 6190  (class class class)co 6978   ∈ cmpo 6980   ↑𝑚 cmap 8208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-1st 7503  df-2nd 7504  df-map 8210 This theorem is referenced by:  clsneiel2  39822  clsneifv4  39824
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