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Theorem clsneiel1 38930
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 38924 . . 3 (𝜑𝐵 ∈ V)
54ancli 538 . 2 (𝜑 → (𝜑𝐵 ∈ V))
6 clsnei.o . . . . . 6 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
7 simpr 471 . . . . . . 7 ((𝜑𝐵 ∈ V) → 𝐵 ∈ V)
8 pwexg 4981 . . . . . . 7 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
97, 8syl 17 . . . . . 6 ((𝜑𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
10 clsnei.f . . . . . 6 𝐹 = (𝒫 𝐵𝑂𝐵)
116, 9, 7, 10fsovfd 38830 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)⟶(𝒫 𝒫 𝐵𝑚 𝐵))
1211ffnd 6185 . . . 4 ((𝜑𝐵 ∈ V) → 𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵))
13 clsnei.p . . . . . 6 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
1413, 1, 7dssmapf1od 38839 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
15 f1of 6279 . . . . 5 (𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵) → 𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)⟶(𝒫 𝐵𝑚 𝒫 𝐵))
1614, 15syl 17 . . . 4 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)⟶(𝒫 𝐵𝑚 𝒫 𝐵))
172breqi 4793 . . . . . 6 (𝐾𝐻𝑁𝐾(𝐹𝐷)𝑁)
183, 17sylib 208 . . . . 5 (𝜑𝐾(𝐹𝐷)𝑁)
1918adantr 466 . . . 4 ((𝜑𝐵 ∈ V) → 𝐾(𝐹𝐷)𝑁)
2012, 16, 19brcoffn 38852 . . 3 ((𝜑𝐵 ∈ V) → (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁))
2120ancli 538 . 2 ((𝜑𝐵 ∈ V) → ((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)))
22 simprl 754 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝐾𝐷(𝐷𝐾))
23 clsneiel.x . . . . 5 (𝜑𝑋𝐵)
2423ad2antrr 705 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝑋𝐵)
25 clsneiel.s . . . . 5 (𝜑𝑆 ∈ 𝒫 𝐵)
2625ad2antrr 705 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝑆 ∈ 𝒫 𝐵)
2713, 1, 22, 24, 26ntrclselnel1 38879 . . 3 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝑋 ∈ (𝐾𝑆) ↔ ¬ 𝑋 ∈ ((𝐷𝐾)‘(𝐵𝑆))))
28 simprr 756 . . . . 5 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝐷𝐾)𝐹𝑁)
29 simplr 752 . . . . . 6 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → 𝐵 ∈ V)
30 difssd 3889 . . . . . 6 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝐵𝑆) ⊆ 𝐵)
3129, 30sselpwd 4942 . . . . 5 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝐵𝑆) ∈ 𝒫 𝐵)
326, 10, 28, 24, 31ntrneiel 38903 . . . 4 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝑋 ∈ ((𝐷𝐾)‘(𝐵𝑆)) ↔ (𝐵𝑆) ∈ (𝑁𝑋)))
3332notbid 307 . . 3 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (¬ 𝑋 ∈ ((𝐷𝐾)‘(𝐵𝑆)) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))
3427, 33bitrd 268 . 2 (((𝜑𝐵 ∈ V) ∧ (𝐾𝐷(𝐷𝐾) ∧ (𝐷𝐾)𝐹𝑁)) → (𝑋 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))
355, 21, 343syl 18 1 (𝜑 → (𝑋 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  cdif 3720  𝒫 cpw 4298   class class class wbr 4787  cmpt 4864  ccom 5254  wf 6026  1-1-ontowf1o 6029  cfv 6030  (class class class)co 6796  cmpt2 6798  𝑚 cmap 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-map 8015
This theorem is referenced by:  clsneiel2  38931  clsneifv4  38933
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