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Theorem ntrneifv1 41155
 Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the function value of 𝐹 is the neighborhood function. (Contributed by RP, 29-May-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
ntrneifv1 (𝜑 → (𝐹𝐼) = 𝑁)
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚   𝜑,𝑖,𝑗,𝑘,𝑙
Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑖,𝑗,𝑘,𝑚,𝑙)   𝐼(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑙)

Proof of Theorem ntrneifv1
StepHypRef Expression
1 ntrnei.r . 2 (𝜑𝐼𝐹𝑁)
2 ntrnei.o . . . . . 6 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
3 ntrnei.f . . . . . 6 𝐹 = (𝒫 𝐵𝑂𝐵)
42, 3, 1ntrneif1o 41151 . . . . 5 (𝜑𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
5 f1ofn 6603 . . . . 5 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → 𝐹 Fn (𝒫 𝐵m 𝒫 𝐵))
64, 5syl 17 . . . 4 (𝜑𝐹 Fn (𝒫 𝐵m 𝒫 𝐵))
72, 3, 1ntrneiiex 41152 . . . 4 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
86, 7jca 515 . . 3 (𝜑 → (𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)))
9 fnfun 6434 . . . . 5 (𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) → Fun 𝐹)
109adantr 484 . . . 4 ((𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → Fun 𝐹)
11 fndm 6436 . . . . . 6 (𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) → dom 𝐹 = (𝒫 𝐵m 𝒫 𝐵))
1211eleq2d 2837 . . . . 5 (𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) → (𝐼 ∈ dom 𝐹𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)))
1312biimpar 481 . . . 4 ((𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → 𝐼 ∈ dom 𝐹)
1410, 13jca 515 . . 3 ((𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (Fun 𝐹𝐼 ∈ dom 𝐹))
15 funbrfvb 6708 . . 3 ((Fun 𝐹𝐼 ∈ dom 𝐹) → ((𝐹𝐼) = 𝑁𝐼𝐹𝑁))
168, 14, 153syl 18 . 2 (𝜑 → ((𝐹𝐼) = 𝑁𝐼𝐹𝑁))
171, 16mpbird 260 1 (𝜑 → (𝐹𝐼) = 𝑁)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3074  Vcvv 3409  𝒫 cpw 4494   class class class wbr 5032   ↦ cmpt 5112  dom cdm 5524  Fun wfun 6329   Fn wfn 6330  –1-1-onto→wf1o 6334  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152   ↑m cmap 8416 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 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-map 8418 This theorem is referenced by:  ntrneiel  41157
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