Theorem ntrneinex 39157

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 39155	. . . 4 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵))
5		f1orel 6359	. . . 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 5563	. . 3 ⊢ ((Rel 𝐹 ∧ 𝐼𝐹𝑁) → 𝑁 ∈ ran 𝐹)
8	6, 3, 7	syl2anc 580	. 2 ⊢ (𝜑 → 𝑁 ∈ ran 𝐹)
9		dff1o2 6361	. . . 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 210	. . 3 ⊢ (𝜑 → (𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ Fun ◡𝐹 ∧ ran 𝐹 = (𝒫 𝒫 𝐵 ↑𝑚 𝐵)))
11	10	simp3d 1175	. 2 ⊢ (𝜑 → ran 𝐹 = (𝒫 𝒫 𝐵 ↑𝑚 𝐵))
12	8, 11	eleqtrd 2880	1 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑𝑚 𝐵))

Syntax hints:   → wi 4   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  {crab 3093  Vcvv 3385  𝒫 cpw 4349   class class class wbr 4843   ↦ cmpt 4922  ^◡ccnv 5311  ran crn 5313  Rel wrel 5317  Fun wfun 6095   Fn wfn 6096  –1-1-onto→wf1o 6100  ‘cfv 6101  (class class class)co 6878   ↦ cmpt2 6880   ↑_𝑚 cmap 8095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-map 8097

This theorem is referenced by:  ntrneifv2  39160  ntrneifv3  39162  ntrneineine0lem  39163  ntrneineine1lem  39164  ntrneiel2  39166  clsneinex  39187  neicvgmex  39197