Theorem ntrneinex 42813

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the neighborhood function 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
ntrneinex	⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚   𝜑,𝑖,𝑗,𝑘,𝑙

Allowed substitution hints:   𝜑(𝑚)   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 ntrneinex

Step	Hyp	Ref	Expression
1		ntrnei.o	. . . . 5 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
2		ntrnei.f	. . . . 5 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
3		ntrnei.r	. . . . 5 ⊢ (𝜑 → 𝐼𝐹𝑁)
4	1, 2, 3	ntrneif1o 42811	. . . 4 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
5		f1orel 6833	. . . 4 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → Rel 𝐹)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → Rel 𝐹)
7		relelrn 5942	. . 3 ⊢ ((Rel 𝐹 ∧ 𝐼𝐹𝑁) → 𝑁 ∈ ran 𝐹)
8	6, 3, 7	syl2anc 584	. 2 ⊢ (𝜑 → 𝑁 ∈ ran 𝐹)
9		dff1o2 6835	. . . 4 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ↔ (𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ Fun ◡𝐹 ∧ ran 𝐹 = (𝒫 𝒫 𝐵 ↑m 𝐵)))
10	4, 9	sylib 217	. . 3 ⊢ (𝜑 → (𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ Fun ◡𝐹 ∧ ran 𝐹 = (𝒫 𝒫 𝐵 ↑m 𝐵)))
11	10	simp3d 1144	. 2 ⊢ (𝜑 → ran 𝐹 = (𝒫 𝒫 𝐵 ↑m 𝐵))
12	8, 11	eleqtrd 2835	1 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474  𝒫 cpw 4601   class class class wbr 5147   ↦ cmpt 5230  ^◡ccnv 5674  ran crn 5676  Rel wrel 5680  Fun wfun 6534   Fn wfn 6535  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407   ↑_m cmap 8816

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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818

This theorem is referenced by:  ntrneifv2  42816  ntrneifv3  42818  ntrneineine0lem  42819  ntrneineine1lem  42820  ntrneiel2  42822  clsneinex  42843  neicvgmex  42853