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 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 7370	. . . . 5 ⊢ (𝑖 = 𝑎 → (𝒫 𝑗 ↑m 𝑖) = (𝒫 𝑗 ↑m 𝑎))
3		rabeq 3419	. . . . . 6 ⊢ (𝑖 = 𝑎 → {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)} = {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)})
4	3	mpteq2dv 5212	. . . . 5 ⊢ (𝑖 = 𝑎 → (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}) = (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}))
5	2, 4	mpteq12dv 5201	. . . 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 4579	. . . . . 6 ⊢ (𝑗 = 𝑏 → 𝒫 𝑗 = 𝒫 𝑏)
7	6	oveq1d 7377	. . . . 5 ⊢ (𝑗 = 𝑏 → (𝒫 𝑗 ↑m 𝑎) = (𝒫 𝑏 ↑m 𝑎))
8		mpteq1 5203	. . . . 5 ⊢ (𝑗 = 𝑏 → (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}) = (𝑙 ∈ 𝑏 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}))
9	7, 8	mpteq12dv 5201	. . . 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 7457	. . 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 2759	. 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 42423	1 ⊢ (𝜑 → 𝐵 ∈ V)

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