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Theorem ntrneicls00 38907
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 38894 . . . . . 6 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
5 elmapi 8047 . . . . . 6 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
64, 5syl 17 . . . . 5 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
71, 2, 3ntrneibex 38891 . . . . . 6 (𝜑𝐵 ∈ V)
8 pwidg 4317 . . . . . 6 (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
97, 8syl 17 . . . . 5 (𝜑𝐵 ∈ 𝒫 𝐵)
106, 9ffvelrnd 6524 . . . 4 (𝜑 → (𝐼𝐵) ∈ 𝒫 𝐵)
1110elpwid 4314 . . 3 (𝜑 → (𝐼𝐵) ⊆ 𝐵)
12 eqss 3759 . . . . 5 ((𝐼𝐵) = 𝐵 ↔ ((𝐼𝐵) ⊆ 𝐵𝐵 ⊆ (𝐼𝐵)))
13 dfss3 3733 . . . . . 6 (𝐵 ⊆ (𝐼𝐵) ↔ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵))
1413anbi2i 732 . . . . 5 (((𝐼𝐵) ⊆ 𝐵𝐵 ⊆ (𝐼𝐵)) ↔ ((𝐼𝐵) ⊆ 𝐵 ∧ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵)))
1512, 14bitri 264 . . . 4 ((𝐼𝐵) = 𝐵 ↔ ((𝐼𝐵) ⊆ 𝐵 ∧ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵)))
1615a1i 11 . . 3 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ ((𝐼𝐵) ⊆ 𝐵 ∧ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵))))
1711, 16mpbirand 531 . 2 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵)))
183adantr 472 . . . 4 ((𝜑𝑥𝐵) → 𝐼𝐹𝑁)
19 simpr 479 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
209adantr 472 . . . 4 ((𝜑𝑥𝐵) → 𝐵 ∈ 𝒫 𝐵)
211, 2, 18, 19, 20ntrneiel 38899 . . 3 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐼𝐵) ↔ 𝐵 ∈ (𝑁𝑥)))
2221ralbidva 3123 . 2 (𝜑 → (∀𝑥𝐵 𝑥 ∈ (𝐼𝐵) ↔ ∀𝑥𝐵 𝐵 ∈ (𝑁𝑥)))
2317, 22bitrd 268 1 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ ∀𝑥𝐵 𝐵 ∈ (𝑁𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  wss 3715  𝒫 cpw 4302   class class class wbr 4804  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6814  cmpt2 6816  𝑚 cmap 8025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027
This theorem is referenced by: (None)
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