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Theorem ntrneifv1 39914
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 39910 . . . . 5 (𝜑𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
5 f1ofn 6484 . . . . 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 39911 . . . 4 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
86, 7jca 512 . . 3 (𝜑 → (𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)))
9 fnfun 6323 . . . . 5 (𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) → Fun 𝐹)
109adantr 481 . . . 4 ((𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → Fun 𝐹)
11 fndm 6325 . . . . . 6 (𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) → dom 𝐹 = (𝒫 𝐵𝑚 𝒫 𝐵))
1211eleq2d 2868 . . . . 5 (𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) → (𝐼 ∈ dom 𝐹𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)))
1312biimpar 478 . . . 4 ((𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → 𝐼 ∈ dom 𝐹)
1410, 13jca 512 . . 3 ((𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (Fun 𝐹𝐼 ∈ dom 𝐹))
15 funbrfvb 6588 . . 3 ((Fun 𝐹𝐼 ∈ dom 𝐹) → ((𝐹𝐼) = 𝑁𝐼𝐹𝑁))
168, 14, 153syl 18 . 2 (𝜑 → ((𝐹𝐼) = 𝑁𝐼𝐹𝑁))
171, 16mpbird 258 1 (𝜑 → (𝐹𝐼) = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  {crab 3109  Vcvv 3437  𝒫 cpw 4453   class class class wbr 4962  cmpt 5041  dom cdm 5443  Fun wfun 6219   Fn wfn 6220  1-1-ontowf1o 6224  cfv 6225  (class class class)co 7016  cmpo 7018  𝑚 cmap 8256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-map 8258
This theorem is referenced by:  ntrneiel  39916
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