Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ntrneibex Structured version   Visualization version   GIF version

Theorem ntrneibex 41653
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 7279 . . . . 5 (𝑖 = 𝑎 → (𝒫 𝑗m 𝑖) = (𝒫 𝑗m 𝑎))
3 rabeq 3417 . . . . . 6 (𝑖 = 𝑎 → {𝑚𝑖𝑙 ∈ (𝑘𝑚)} = {𝑚𝑎𝑙 ∈ (𝑘𝑚)})
43mpteq2dv 5181 . . . . 5 (𝑖 = 𝑎 → (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)}) = (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)}))
52, 4mpteq12dv 5170 . . . 4 (𝑖 = 𝑎 → (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})) = (𝑘 ∈ (𝒫 𝑗m 𝑎) ↦ (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)})))
6 pweq 4555 . . . . . 6 (𝑗 = 𝑏 → 𝒫 𝑗 = 𝒫 𝑏)
76oveq1d 7286 . . . . 5 (𝑗 = 𝑏 → (𝒫 𝑗m 𝑎) = (𝒫 𝑏m 𝑎))
8 mpteq1 5172 . . . . 5 (𝑗 = 𝑏 → (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)}) = (𝑙𝑏 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)}))
97, 8mpteq12dv 5170 . . . 4 (𝑗 = 𝑏 → (𝑘 ∈ (𝒫 𝑗m 𝑎) ↦ (𝑙𝑗 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)})) = (𝑘 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑙𝑏 ↦ {𝑚𝑎𝑙 ∈ (𝑘𝑚)})))
105, 9cbvmpov 7364 . . 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 2768 . 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 41609 1 (𝜑𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2110  {crab 3070  Vcvv 3431  𝒫 cpw 4539   class class class wbr 5079  cmpt 5162  cfv 6432  (class class class)co 7271  cmpo 7273  m cmap 8598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-xp 5596  df-rel 5597  df-dm 5600  df-iota 6390  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276
This theorem is referenced by:  ntrneircomplex  41654  ntrneif1o  41655  ntrneicnv  41658  ntrneiel  41661  ntrneicls00  41669  ntrneik3  41676  ntrneix3  41677  ntrneik13  41678  ntrneix13  41679
  Copyright terms: Public domain W3C validator