Theorem ntrneibex 38371

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 6658	. . . . 5 ⊢ (𝑖 = 𝑎 → (𝒫 𝑗 ↑𝑚 𝑖) = (𝒫 𝑗 ↑𝑚 𝑎))
3		rabeq 3192	. . . . . 6 ⊢ (𝑖 = 𝑎 → {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)} = {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)})
4	3	mpteq2dv 4745	. . . . 5 ⊢ (𝑖 = 𝑎 → (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}) = (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}))
5	2, 4	mpteq12dv 4733	. . . 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 4161	. . . . . 6 ⊢ (𝑗 = 𝑏 → 𝒫 𝑗 = 𝒫 𝑏)
7	6	oveq1d 6665	. . . . 5 ⊢ (𝑗 = 𝑏 → (𝒫 𝑗 ↑𝑚 𝑎) = (𝒫 𝑏 ↑𝑚 𝑎))
8		mpteq1 4737	. . . . 5 ⊢ (𝑗 = 𝑏 → (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}) = (𝑙 ∈ 𝑏 ↦ {𝑚 ∈ 𝑎 ∣ 𝑙 ∈ (𝑘‘𝑚)}))
9	7, 8	mpteq12dv 4733	. . . 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	cbvmpt2v 6735	. . 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 2644	. 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 38327	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200  𝒫 cpw 4158   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  ntrneircomplex  38372  ntrneif1o  38373  ntrneicnv  38376  ntrneiel  38379  ntrneicls00  38387  ntrneik3  38394  ntrneix3  38395  ntrneik13  38396  ntrneix13  38397