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Theorem ntrneibex 40586
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 7138 . . . . 5 (𝑖 = 𝑎 → (𝒫 𝑗m 𝑖) = (𝒫 𝑗m 𝑎))
3 rabeq 3460 . . . . . 6 (𝑖 = 𝑎 → {𝑚𝑖𝑙 ∈ (𝑘𝑚)} = {𝑚𝑎𝑙 ∈ (𝑘𝑚)})
43mpteq2dv 5135 . . . . 5 (𝑖 = 𝑎 → (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)}) = (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)}))
52, 4mpteq12dv 5124 . . . 4 (𝑖 = 𝑎 → (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})) = (𝑘 ∈ (𝒫 𝑗m 𝑎) ↦ (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)})))
6 pweq 4528 . . . . . 6 (𝑗 = 𝑏 → 𝒫 𝑗 = 𝒫 𝑏)
76oveq1d 7145 . . . . 5 (𝑗 = 𝑏 → (𝒫 𝑗m 𝑎) = (𝒫 𝑏m 𝑎))
8 mpteq1 5127 . . . . 5 (𝑗 = 𝑏 → (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)}) = (𝑙𝑏 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)}))
97, 8mpteq12dv 5124 . . . 4 (𝑗 = 𝑏 → (𝑘 ∈ (𝒫 𝑗m 𝑎) ↦ (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)})) = (𝑘 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑙𝑏 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)})))
105, 9cbvmpov 7223 . . 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 2844 . 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 40542 1 (𝜑𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  {crab 3130  Vcvv 3471  𝒫 cpw 4512   class class class wbr 5039  cmpt 5119  cfv 6328  (class class class)co 7130  cmpo 7132  m cmap 8381
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-xp 5534  df-rel 5535  df-dm 5538  df-iota 6287  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135
This theorem is referenced by:  ntrneircomplex  40587  ntrneif1o  40588  ntrneicnv  40591  ntrneiel  40594  ntrneicls00  40602  ntrneik3  40609  ntrneix3  40610  ntrneik13  40611  ntrneix13  40612
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