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Theorem ntrneifv1 37894
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 37890 . . . . 5 (𝜑𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
5 f1ofn 6100 . . . . 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 37891 . . . 4 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
86, 7jca 554 . . 3 (𝜑 → (𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)))
9 fnfun 5951 . . . . 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 5953 . . . . . 6 (𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) → dom 𝐹 = (𝒫 𝐵𝑚 𝒫 𝐵))
1211eleq2d 2684 . . . . 5 (𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) → (𝐼 ∈ dom 𝐹𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)))
1312biimpar 502 . . . 4 ((𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → 𝐼 ∈ dom 𝐹)
1410, 13jca 554 . . 3 ((𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (Fun 𝐹𝐼 ∈ dom 𝐹))
15 funbrfvb 6200 . . 3 ((Fun 𝐹𝐼 ∈ dom 𝐹) → ((𝐹𝐼) = 𝑁𝐼𝐹𝑁))
168, 14, 153syl 18 . 2 (𝜑 → ((𝐹𝐼) = 𝑁𝐼𝐹𝑁))
171, 16mpbird 247 1 (𝜑 → (𝐹𝐼) = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3189  𝒫 cpw 4135   class class class wbr 4618  cmpt 4678  dom cdm 5079  Fun wfun 5846   Fn wfn 5847  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  cmpt2 6612  𝑚 cmap 7809
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-map 7811
This theorem is referenced by:  ntrneiel  37896
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