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Theorem ntrneicls00 37890
 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 37877 . . . . . 6 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
5 elmapi 7826 . . . . . 6 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
64, 5syl 17 . . . . 5 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
71, 2, 3ntrneibex 37874 . . . . . 6 (𝜑𝐵 ∈ V)
8 pwidg 4146 . . . . . 6 (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
97, 8syl 17 . . . . 5 (𝜑𝐵 ∈ 𝒫 𝐵)
106, 9ffvelrnd 6318 . . . 4 (𝜑 → (𝐼𝐵) ∈ 𝒫 𝐵)
1110elpwid 4143 . . 3 (𝜑 → (𝐼𝐵) ⊆ 𝐵)
12 eqss 3599 . . . . 5 ((𝐼𝐵) = 𝐵 ↔ ((𝐼𝐵) ⊆ 𝐵𝐵 ⊆ (𝐼𝐵)))
13 dfss3 3574 . . . . . 6 (𝐵 ⊆ (𝐼𝐵) ↔ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵))
1413anbi2i 729 . . . . 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 530 . 2 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ ∀𝑥𝐵 𝑥 ∈ (𝐼𝐵)))
183adantr 481 . . . 4 ((𝜑𝑥𝐵) → 𝐼𝐹𝑁)
19 simpr 477 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
209adantr 481 . . . 4 ((𝜑𝑥𝐵) → 𝐵 ∈ 𝒫 𝐵)
211, 2, 18, 19, 20ntrneiel 37882 . . 3 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐼𝐵) ↔ 𝐵 ∈ (𝑁𝑥)))
2221ralbidva 2979 . 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 384   = wceq 1480   ∈ wcel 1987  ∀wral 2907  {crab 2911  Vcvv 3186   ⊆ wss 3556  𝒫 cpw 4132   class class class wbr 4615   ↦ cmpt 4675  ⟶wf 5845  ‘cfv 5849  (class class class)co 6607   ↦ cmpt2 6609   ↑𝑚 cmap 7805 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-map 7807 This theorem is referenced by: (None)
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