Theorem ntrneinex 45021

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 45019	. . . 4 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
5		f1orel 6815	. . . 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 5923	. . 3 ⊢ ((Rel 𝐹 ∧ 𝐼𝐹𝑁) → 𝑁 ∈ ran 𝐹)
8	6, 3, 7	syl2anc 596	. 2 ⊢ (𝜑 → 𝑁 ∈ ran 𝐹)
9		dff1o2 6818	. . . 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 2862	1 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3412  Vcvv 3450  𝒫 cpw 4556   class class class wbr 5102   ↦ cmpt 5185  ^◡ccnv 5646  ran crn 5648  Rel wrel 5652  Fun wfun 6521   Fn wfn 6522  –1-1-onto→wf1o 6526  ‘cfv 6527  (class class class)co 7408   ∈ cmpo 7410   ↑_m cmap 8825

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 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-map 8827

This theorem is used by:  ntrneifv2  45024  ntrneifv3  45026  ntrneineine0lem  45027  ntrneineine1lem  45028  ntrneiel2  45030  clsneinex  45051  neicvgmex  45061