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Theorem ntrneifv1 38903
 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 38899 . . . . 5 (𝜑𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
5 f1ofn 6279 . . . . 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 38900 . . . 4 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
86, 7jca 501 . . 3 (𝜑 → (𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)))
9 fnfun 6128 . . . . 5 (𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) → Fun 𝐹)
109adantr 466 . . . 4 ((𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → Fun 𝐹)
11 fndm 6130 . . . . . 6 (𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) → dom 𝐹 = (𝒫 𝐵𝑚 𝒫 𝐵))
1211eleq2d 2836 . . . . 5 (𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) → (𝐼 ∈ dom 𝐹𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)))
1312biimpar 463 . . . 4 ((𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → 𝐼 ∈ dom 𝐹)
1410, 13jca 501 . . 3 ((𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (Fun 𝐹𝐼 ∈ dom 𝐹))
15 funbrfvb 6379 . . 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 382   = wceq 1631   ∈ wcel 2145  {crab 3065  Vcvv 3351  𝒫 cpw 4297   class class class wbr 4786   ↦ cmpt 4863  dom cdm 5249  Fun wfun 6025   Fn wfn 6026  –1-1-onto→wf1o 6030  ‘cfv 6031  (class class class)co 6793   ↦ cmpt2 6795   ↑𝑚 cmap 8009 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011 This theorem is referenced by:  ntrneiel  38905
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