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Theorem ntrneicls00 40790
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 2-Jun-2021.)
Hypotheses
Ref Expression
ntrnei.o 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
ntrnei.f 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r (𝜑𝐼𝐹𝑁)
Assertion
Ref Expression
ntrneicls00 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ ∀𝑥𝐵 𝐵 ∈ (𝑁𝑥)))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑥   𝑘,𝐼,𝑙,𝑚,𝑥   𝜑,𝑖,𝑗,𝑘,𝑙,𝑥
Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)   𝐼(𝑖,𝑗)   𝑁(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)

Proof of Theorem ntrneicls00
StepHypRef Expression
1 ntrnei.o . . . . . . 7 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
2 ntrnei.f . . . . . . 7 𝐹 = (𝒫 𝐵𝑂𝐵)
3 ntrnei.r . . . . . . 7 (𝜑𝐼𝐹𝑁)
41, 2, 3ntrneiiex 40777 . . . . . 6 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
5 elmapi 8411 . . . . . 6 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
64, 5syl 17 . . . . 5 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
71, 2, 3ntrneibex 40774 . . . . . 6 (𝜑𝐵 ∈ V)
8 pwidg 4519 . . . . . 6 (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
97, 8syl 17 . . . . 5 (𝜑𝐵 ∈ 𝒫 𝐵)
106, 9ffvelrnd 6829 . . . 4 (𝜑 → (𝐼𝐵) ∈ 𝒫 𝐵)
1110elpwid 4508 . . 3 (𝜑 → (𝐼𝐵) ⊆ 𝐵)
12 eqss 3930 . . . . 5 ((𝐼𝐵) = 𝐵 ↔ ((𝐼𝐵) ⊆ 𝐵𝐵 ⊆ (𝐼𝐵)))
13 dfss3 3903 . . . . . 6 (𝐵 ⊆ (𝐼𝐵) ↔ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵))
1413anbi2i 625 . . . . 5 (((𝐼𝐵) ⊆ 𝐵𝐵 ⊆ (𝐼𝐵)) ↔ ((𝐼𝐵) ⊆ 𝐵 ∧ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵)))
1512, 14bitri 278 . . . 4 ((𝐼𝐵) = 𝐵 ↔ ((𝐼𝐵) ⊆ 𝐵 ∧ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵)))
1615a1i 11 . . 3 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ ((𝐼𝐵) ⊆ 𝐵 ∧ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵))))
1711, 16mpbirand 706 . 2 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵)))
183adantr 484 . . . 4 ((𝜑𝑥𝐵) → 𝐼𝐹𝑁)
19 simpr 488 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
209adantr 484 . . . 4 ((𝜑𝑥𝐵) → 𝐵 ∈ 𝒫 𝐵)
211, 2, 18, 19, 20ntrneiel 40782 . . 3 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐼𝐵) ↔ 𝐵 ∈ (𝑁𝑥)))
2221ralbidva 3161 . 2 (𝜑 → (∀𝑥𝐵 𝑥 ∈ (𝐼𝐵) ↔ ∀𝑥𝐵 𝐵 ∈ (𝑁𝑥)))
2317, 22bitrd 282 1 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ ∀𝑥𝐵 𝐵 ∈ (𝑁𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wral 3106  {crab 3110  Vcvv 3441  wss 3881  𝒫 cpw 4497   class class class wbr 5030  cmpt 5110  wf 6320  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: (None)
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