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Theorem ntrneibex 42467
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the base set exists. (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
ntrneibex (𝜑𝐵 ∈ V)
Distinct variable groups:   𝑖,𝑗,𝑘   𝑖,𝑙,𝑗   𝑖,𝑚,𝑗
Allowed substitution hints:   𝜑(𝑖,𝑗,𝑘,𝑚,𝑙)   𝐵(𝑖,𝑗,𝑘,𝑚,𝑙)   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 ntrneibex
Dummy variables 𝑏 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ntrnei.o . . 3 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
2 oveq2 7370 . . . . 5 (𝑖 = 𝑎 → (𝒫 𝑗m 𝑖) = (𝒫 𝑗m 𝑎))
3 rabeq 3419 . . . . . 6 (𝑖 = 𝑎 → {𝑚𝑖𝑙 ∈ (𝑘𝑚)} = {𝑚𝑎𝑙 ∈ (𝑘𝑚)})
43mpteq2dv 5212 . . . . 5 (𝑖 = 𝑎 → (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)}) = (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)}))
52, 4mpteq12dv 5201 . . . 4 (𝑖 = 𝑎 → (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})) = (𝑘 ∈ (𝒫 𝑗m 𝑎) ↦ (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)})))
6 pweq 4579 . . . . . 6 (𝑗 = 𝑏 → 𝒫 𝑗 = 𝒫 𝑏)
76oveq1d 7377 . . . . 5 (𝑗 = 𝑏 → (𝒫 𝑗m 𝑎) = (𝒫 𝑏m 𝑎))
8 mpteq1 5203 . . . . 5 (𝑗 = 𝑏 → (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)}) = (𝑙𝑏 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)}))
97, 8mpteq12dv 5201 . . . 4 (𝑗 = 𝑏 → (𝑘 ∈ (𝒫 𝑗m 𝑎) ↦ (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)})) = (𝑘 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑙𝑏 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)})))
105, 9cbvmpov 7457 . . 3 (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)}))) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑘 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑙𝑏 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)})))
111, 10eqtri 2759 . 2 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑘 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑙𝑏 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)})))
12 ntrnei.r . 2 (𝜑𝐼𝐹𝑁)
13 ntrnei.f . . 3 𝐹 = (𝒫 𝐵𝑂𝐵)
1413a1i 11 . 2 (𝜑𝐹 = (𝒫 𝐵𝑂𝐵))
1511, 12, 14brovmptimex2 42423 1 (𝜑𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {crab 3405  Vcvv 3446  𝒫 cpw 4565   class class class wbr 5110  cmpt 5193  cfv 6501  (class class class)co 7362  cmpo 7364  m cmap 8772
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-xp 5644  df-rel 5645  df-dm 5648  df-iota 6453  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367
This theorem is referenced by:  ntrneircomplex  42468  ntrneif1o  42469  ntrneicnv  42472  ntrneiel  42475  ntrneicls00  42483  ntrneik3  42490  ntrneix3  42491  ntrneik13  42492  ntrneix13  42493
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