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Theorem ntrneibex 42824
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 7417 . . . . 5 (𝑖 = 𝑎 → (𝒫 𝑗m 𝑖) = (𝒫 𝑗m 𝑎))
3 rabeq 3447 . . . . . 6 (𝑖 = 𝑎 → {𝑚𝑖𝑙 ∈ (𝑘𝑚)} = {𝑚𝑎𝑙 ∈ (𝑘𝑚)})
43mpteq2dv 5251 . . . . 5 (𝑖 = 𝑎 → (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)}) = (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)}))
52, 4mpteq12dv 5240 . . . 4 (𝑖 = 𝑎 → (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})) = (𝑘 ∈ (𝒫 𝑗m 𝑎) ↦ (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)})))
6 pweq 4617 . . . . . 6 (𝑗 = 𝑏 → 𝒫 𝑗 = 𝒫 𝑏)
76oveq1d 7424 . . . . 5 (𝑗 = 𝑏 → (𝒫 𝑗m 𝑎) = (𝒫 𝑏m 𝑎))
8 mpteq1 5242 . . . . 5 (𝑗 = 𝑏 → (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)}) = (𝑙𝑏 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)}))
97, 8mpteq12dv 5240 . . . 4 (𝑗 = 𝑏 → (𝑘 ∈ (𝒫 𝑗m 𝑎) ↦ (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)})) = (𝑘 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑙𝑏 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)})))
105, 9cbvmpov 7504 . . 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 2761 . 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 42780 1 (𝜑𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  𝒫 cpw 4603   class class class wbr 5149  cmpt 5232  cfv 6544  (class class class)co 7409  cmpo 7411  m cmap 8820
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5683  df-rel 5684  df-dm 5687  df-iota 6496  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414
This theorem is referenced by:  ntrneircomplex  42825  ntrneif1o  42826  ntrneicnv  42829  ntrneiel  42832  ntrneicls00  42840  ntrneik3  42847  ntrneix3  42848  ntrneik13  42849  ntrneix13  42850
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