Theorem ntrneinex 44904

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 44902	. . . 4 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
5		f1orel 6824	. . . 4 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → Rel 𝐹)
6	4, 5	syl 18	. . 3 ⊢ (𝜑 → Rel 𝐹)
7		relelrn 5933	. . 3 ⊢ ((Rel 𝐹 ∧ 𝐼𝐹𝑁) → 𝑁 ∈ ran 𝐹)
8	6, 3, 7	syl2anc 596	. 2 ⊢ (𝜑 → 𝑁 ∈ ran 𝐹)
9		dff1o2 6827	. . . 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 221	. . 3 ⊢ (𝜑 → (𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ Fun ◡𝐹 ∧ ran 𝐹 = (𝒫 𝒫 𝐵 ↑m 𝐵)))
11	10	simp3d 1162	. 2 ⊢ (𝜑 → ran 𝐹 = (𝒫 𝒫 𝐵 ↑m 𝐵))
12	8, 11	eleqtrd 2864	1 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453  𝒫 cpw 4560   class class class wbr 5107   ↦ cmpt 5190  ^◡ccnv 5658  ran crn 5660  Rel wrel 5664  Fun wfun 6531   Fn wfn 6532  –1-1-onto→wf1o 6536  ‘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-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  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-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-map 8831

This theorem is used by:  ntrneifv2  44907  ntrneifv3  44909  ntrneineine0lem  44910  ntrneineine1lem  44911  ntrneiel2  44913  clsneinex  44934  neicvgmex  44944