Theorem ntrneineine1lem 43137

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at not all subsets are (pseudo-)neighborboods hold equally. (Contributed by RP, 1-Jun-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)
ntrnei.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
ntrneineine1lem	⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼‘𝑠) ↔ (𝑁‘𝑋) ≠ 𝒫 𝐵))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠   𝑘,𝐼,𝑙,𝑚   𝑁,𝑠   𝑋,𝑙,𝑚,𝑠   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠

Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝐼(𝑖,𝑗,𝑠)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝑋(𝑖,𝑗,𝑘)

Proof of Theorem ntrneineine1lem

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	. . . . . 6 ⊢ (𝜑 → 𝐼𝐹𝑁)
4	3	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
5		ntrnei.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6	5	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵)
7		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
8	1, 2, 4, 6, 7	ntrneiel 43134	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑋)))
9	8	notbid 317	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (¬ 𝑋 ∈ (𝐼‘𝑠) ↔ ¬ 𝑠 ∈ (𝑁‘𝑋)))
10	9	rexbidva 3174	. 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁‘𝑋)))
11	1, 2, 3	ntrneinex 43130	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
12		elmapi 8845	. . . . . . 7 ⊢ (𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
13	11, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
14	13, 5	ffvelcdmd 7086	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵)
15	14	elpwid 4610	. . . 4 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵)
16		biortn 934	. . . 4 ⊢ ((𝑁‘𝑋) ⊆ 𝒫 𝐵 → (¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋) ↔ (¬ (𝑁‘𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋))))
17	15, 16	syl 17	. . 3 ⊢ (𝜑 → (¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋) ↔ (¬ (𝑁‘𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋))))
18		df-rex 3069	. . . 4 ⊢ (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁‘𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁‘𝑋)))
19		nss 4045	. . . 4 ⊢ (¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁‘𝑋)))
20	18, 19	bitr4i 277	. . 3 ⊢ (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁‘𝑋) ↔ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋))
21		df-ne 2939	. . . 4 ⊢ ((𝑁‘𝑋) ≠ 𝒫 𝐵 ↔ ¬ (𝑁‘𝑋) = 𝒫 𝐵)
22		ianor 978	. . . . 5 ⊢ (¬ ((𝑁‘𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁‘𝑋)) ↔ (¬ (𝑁‘𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋)))
23		eqss 3996	. . . . 5 ⊢ ((𝑁‘𝑋) = 𝒫 𝐵 ↔ ((𝑁‘𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁‘𝑋)))
24	22, 23	xchnxbir 332	. . . 4 ⊢ (¬ (𝑁‘𝑋) = 𝒫 𝐵 ↔ (¬ (𝑁‘𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋)))
25	21, 24	bitri 274	. . 3 ⊢ ((𝑁‘𝑋) ≠ 𝒫 𝐵 ↔ (¬ (𝑁‘𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋)))
26	17, 20, 25	3bitr4g 313	. 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁‘𝑋) ↔ (𝑁‘𝑋) ≠ 𝒫 𝐵))
27	10, 26	bitrd 278	1 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼‘𝑠) ↔ (𝑁‘𝑋) ≠ 𝒫 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   = wceq 1539  ∃wex 1779   ∈ wcel 2104   ≠ wne 2938  ∃wrex 3068  {crab 3430  Vcvv 3472   ⊆ wss 3947  𝒫 cpw 4601   class class class wbr 5147   ↦ cmpt 5230  ⟶wf 6538  ‘cfv 6542  (class class class)co 7411   ∈ cmpo 7413   ↑_m cmap 8822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824

This theorem is referenced by:  ntrneineine1  43141