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Theorem ntrneicls11 38214
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the interior 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
ntrneicls11 (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥𝐵 ¬ ∅ ∈ (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 ntrneicls11
StepHypRef Expression
1 ntrnei.o . . . . . . . . 9 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
2 ntrnei.f . . . . . . . . 9 𝐹 = (𝒫 𝐵𝑂𝐵)
3 ntrnei.r . . . . . . . . 9 (𝜑𝐼𝐹𝑁)
41, 2, 3ntrneiiex 38200 . . . . . . . 8 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
5 elmapi 7876 . . . . . . . 8 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
64, 5syl 17 . . . . . . 7 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
7 0elpw 4832 . . . . . . . 8 ∅ ∈ 𝒫 𝐵
87a1i 11 . . . . . . 7 (𝜑 → ∅ ∈ 𝒫 𝐵)
96, 8ffvelrnd 6358 . . . . . 6 (𝜑 → (𝐼‘∅) ∈ 𝒫 𝐵)
109elpwid 4168 . . . . 5 (𝜑 → (𝐼‘∅) ⊆ 𝐵)
11 reldisj 4018 . . . . 5 ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵𝐵)))
1210, 11syl 17 . . . 4 (𝜑 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵𝐵)))
1312bicomd 213 . . 3 (𝜑 → ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ ((𝐼‘∅) ∩ 𝐵) = ∅))
14 difid 3946 . . . . 5 (𝐵𝐵) = ∅
1514sseq2i 3628 . . . 4 ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ (𝐼‘∅) ⊆ ∅)
16 ss0b 3971 . . . 4 ((𝐼‘∅) ⊆ ∅ ↔ (𝐼‘∅) = ∅)
1715, 16bitri 264 . . 3 ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ (𝐼‘∅) = ∅)
18 disjr 4016 . . 3 (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥 ∈ (𝐼‘∅))
1913, 17, 183bitr3g 302 . 2 (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥 ∈ (𝐼‘∅)))
203adantr 481 . . . . 5 ((𝜑𝑥𝐵) → 𝐼𝐹𝑁)
21 simpr 477 . . . . 5 ((𝜑𝑥𝐵) → 𝑥𝐵)
227a1i 11 . . . . 5 ((𝜑𝑥𝐵) → ∅ ∈ 𝒫 𝐵)
231, 2, 20, 21, 22ntrneiel 38205 . . . 4 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐼‘∅) ↔ ∅ ∈ (𝑁𝑥)))
2423notbid 308 . . 3 ((𝜑𝑥𝐵) → (¬ 𝑥 ∈ (𝐼‘∅) ↔ ¬ ∅ ∈ (𝑁𝑥)))
2524ralbidva 2984 . 2 (𝜑 → (∀𝑥𝐵 ¬ 𝑥 ∈ (𝐼‘∅) ↔ ∀𝑥𝐵 ¬ ∅ ∈ (𝑁𝑥)))
2619, 25bitrd 268 1 (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥𝐵 ¬ ∅ ∈ (𝑁𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wral 2911  {crab 2915  Vcvv 3198  cdif 3569  cin 3571  wss 3572  c0 3913  𝒫 cpw 4156   class class class wbr 4651  cmpt 4727  wf 5882  cfv 5886  (class class class)co 6647  cmpt2 6649  𝑚 cmap 7854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-map 7856
This theorem is referenced by: (None)
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