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Theorem ntrneifv1 42358
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 42354 . . . . 5 (𝜑𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
5 f1ofn 6786 . . . . 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 42355 . . . 4 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
86, 7jca 513 . . 3 (𝜑 → (𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)))
9 fnfun 6603 . . . . 5 (𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) → Fun 𝐹)
109adantr 482 . . . 4 ((𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → Fun 𝐹)
11 fndm 6606 . . . . . 6 (𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) → dom 𝐹 = (𝒫 𝐵m 𝒫 𝐵))
1211eleq2d 2824 . . . . 5 (𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) → (𝐼 ∈ dom 𝐹𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)))
1312biimpar 479 . . . 4 ((𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → 𝐼 ∈ dom 𝐹)
1410, 13jca 513 . . 3 ((𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (Fun 𝐹𝐼 ∈ dom 𝐹))
15 funbrfvb 6898 . . 3 ((Fun 𝐹𝐼 ∈ dom 𝐹) → ((𝐹𝐼) = 𝑁𝐼𝐹𝑁))
168, 14, 153syl 18 . 2 (𝜑 → ((𝐹𝐼) = 𝑁𝐼𝐹𝑁))
171, 16mpbird 257 1 (𝜑 → (𝐹𝐼) = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {crab 3408  Vcvv 3446  𝒫 cpw 4561   class class class wbr 5106  cmpt 5189  dom cdm 5634  Fun wfun 6491   Fn wfn 6492  1-1-ontowf1o 6496  cfv 6497  (class class class)co 7358  cmpo 7360  m cmap 8766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8768
This theorem is referenced by:  ntrneiel  42360
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