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Theorem ntrneicls00 44079
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 44066 . . . . . 6 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
5 elmapi 8888 . . . . . 6 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
64, 5syl 17 . . . . 5 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
71, 2, 3ntrneibex 44063 . . . . . 6 (𝜑𝐵 ∈ V)
8 pwidg 4625 . . . . . 6 (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
97, 8syl 17 . . . . 5 (𝜑𝐵 ∈ 𝒫 𝐵)
106, 9ffvelcdmd 7105 . . . 4 (𝜑 → (𝐼𝐵) ∈ 𝒫 𝐵)
1110elpwid 4614 . . 3 (𝜑 → (𝐼𝐵) ⊆ 𝐵)
12 eqss 4011 . . . . 5 ((𝐼𝐵) = 𝐵 ↔ ((𝐼𝐵) ⊆ 𝐵𝐵 ⊆ (𝐼𝐵)))
13 dfss3 3984 . . . . . 6 (𝐵 ⊆ (𝐼𝐵) ↔ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵))
1413anbi2i 623 . . . . 5 (((𝐼𝐵) ⊆ 𝐵𝐵 ⊆ (𝐼𝐵)) ↔ ((𝐼𝐵) ⊆ 𝐵 ∧ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵)))
1512, 14bitri 275 . . . 4 ((𝐼𝐵) = 𝐵 ↔ ((𝐼𝐵) ⊆ 𝐵 ∧ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵)))
1615a1i 11 . . 3 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ ((𝐼𝐵) ⊆ 𝐵 ∧ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵))))
1711, 16mpbirand 707 . 2 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵)))
183adantr 480 . . . 4 ((𝜑𝑥𝐵) → 𝐼𝐹𝑁)
19 simpr 484 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
209adantr 480 . . . 4 ((𝜑𝑥𝐵) → 𝐵 ∈ 𝒫 𝐵)
211, 2, 18, 19, 20ntrneiel 44071 . . 3 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐼𝐵) ↔ 𝐵 ∈ (𝑁𝑥)))
2221ralbidva 3174 . 2 (𝜑 → (∀𝑥𝐵 𝑥 ∈ (𝐼𝐵) ↔ ∀𝑥𝐵 𝐵 ∈ (𝑁𝑥)))
2317, 22bitrd 279 1 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ ∀𝑥𝐵 𝐵 ∈ (𝑁𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  wss 3963  𝒫 cpw 4605   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by: (None)
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