Theorem ntrneibex 41572

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 7263	. . . . 5 ⊢ (𝑖 = 𝑎 → (𝒫 𝑗 ↑m 𝑖) = (𝒫 𝑗 ↑m 𝑎))
3		rabeq 3408	. . . . . 6 ⊢ (𝑖 = 𝑎 → {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)} = {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)})
4	3	mpteq2dv 5172	. . . . 5 ⊢ (𝑖 = 𝑎 → (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}) = (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}))
5	2, 4	mpteq12dv 5161	. . . 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 4546	. . . . . 6 ⊢ (𝑗 = 𝑏 → 𝒫 𝑗 = 𝒫 𝑏)
7	6	oveq1d 7270	. . . . 5 ⊢ (𝑗 = 𝑏 → (𝒫 𝑗 ↑m 𝑎) = (𝒫 𝑏 ↑m 𝑎))
8		mpteq1 5163	. . . . 5 ⊢ (𝑗 = 𝑏 → (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}) = (𝑙 ∈ 𝑏 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}))
9	7, 8	mpteq12dv 5161	. . . 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 7348	. . 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 2766	. 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 41528	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422  𝒫 cpw 4530   class class class wbr 5070   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257   ↑_m cmap 8573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-xp 5586  df-rel 5587  df-dm 5590  df-iota 6376  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260

This theorem is referenced by:  ntrneircomplex  41573  ntrneif1o  41574  ntrneicnv  41577  ntrneiel  41580  ntrneicls00  41588  ntrneik3  41595  ntrneix3  41596  ntrneik13  41597  ntrneix13  41598