Theorem ntrneibex 40776

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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
ntrnei.f	⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r	⊢ (𝜑 → 𝐼𝐹𝑁)

Assertion

Ref	Expression
ntrneibex	⊢ (𝜑 → 𝐵 ∈ 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.

Step	Hyp	Ref	Expression
1		ntrnei.o	. . 3 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
2		oveq2 7143	. . . . 5 ⊢ (𝑖 = 𝑎 → (𝒫 𝑗 ↑m 𝑖) = (𝒫 𝑗 ↑m 𝑎))
3		rabeq 3431	. . . . . 6 ⊢ (𝑖 = 𝑎 → {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)} = {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)})
4	3	mpteq2dv 5126	. . . . 5 ⊢ (𝑖 = 𝑎 → (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}) = (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}))
5	2, 4	mpteq12dv 5115	. . . 4 ⊢ (𝑖 = 𝑎 → (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})) = (𝑘 ∈ (𝒫 𝑗 ↑m 𝑎) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
6		pweq 4513	. . . . . 6 ⊢ (𝑗 = 𝑏 → 𝒫 𝑗 = 𝒫 𝑏)
7	6	oveq1d 7150	. . . . 5 ⊢ (𝑗 = 𝑏 → (𝒫 𝑗 ↑m 𝑎) = (𝒫 𝑏 ↑m 𝑎))
8		mpteq1 5118	. . . . 5 ⊢ (𝑗 = 𝑏 → (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}) = (𝑙 ∈ 𝑏 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}))
9	7, 8	mpteq12dv 5115	. . . 4 ⊢ (𝑗 = 𝑏 → (𝑘 ∈ (𝒫 𝑗 ↑m 𝑎) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)})) = (𝑘 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑙 ∈ 𝑏 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
10	5, 9	cbvmpov 7228	. . 3 ⊢ (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑘 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑙 ∈ 𝑏 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
11	1, 10	eqtri 2821	. 2 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑘 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑙 ∈ 𝑏 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
12		ntrnei.r	. 2 ⊢ (𝜑 → 𝐼𝐹𝑁)
13		ntrnei.f	. . 3 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
14	13	a1i 11	. 2 ⊢ (𝜑 → 𝐹 = (𝒫 𝐵𝑂𝐵))
15	11, 12, 14	brovmptimex2 40732	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441  𝒫 cpw 4497   class class class wbr 5030   ↦ cmpt 5110  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   ↑_m cmap 8389

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5525  df-rel 5526  df-dm 5529  df-iota 6283  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140

This theorem is referenced by:  ntrneircomplex  40777  ntrneif1o  40778  ntrneicnv  40781  ntrneiel  40784  ntrneicls00  40792  ntrneik3  40799  ntrneix3  40800  ntrneik13  40801  ntrneix13  40802