Theorem ntrneibex 44900

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 7424	. . . . 5 ⊢ (𝑖 = 𝑎 → (𝒫 𝑗 ↑m 𝑖) = (𝒫 𝑗 ↑m 𝑎))
3		rabeq 3428	. . . . . 6 ⊢ (𝑖 = 𝑎 → {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)} = {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)})
4	3	mpteq2dv 5203	. . . . 5 ⊢ (𝑖 = 𝑎 → (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}) = (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}))
5	2, 4	mpteq12dv 5196	. . . 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 4574	. . . . . 6 ⊢ (𝑗 = 𝑏 → 𝒫 𝑗 = 𝒫 𝑏)
7	6	oveq1d 7431	. . . . 5 ⊢ (𝑗 = 𝑏 → (𝒫 𝑗 ↑m 𝑎) = (𝒫 𝑏 ↑m 𝑎))
8		mpteq1 5198	. . . . 5 ⊢ (𝑗 = 𝑏 → (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}) = (𝑙 ∈ 𝑏 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}))
9	7, 8	mpteq12dv 5196	. . . 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 7511	. . 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 2785	. 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 44856	1 ⊢ (𝜑 → 𝐵 ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453  𝒫 cpw 4560   class class class wbr 5107   ↦ cmpt 5190  ‘cfv 6537  (class class class)co 7416   ∈ cmpo 7418   ↑_m cmap 8829

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-xp 5665  df-rel 5666  df-dm 5669  df-iota 6493  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421

This theorem is used by:  ntrneircomplex  44901  ntrneif1o  44902  ntrneicnv  44905  ntrneiel  44908  ntrneicls00  44916  ntrneik3  44923  ntrneix3  44924  ntrneik13  44925  ntrneix13  44926