Theorem ntrneineine1lem 44073

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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
5		ntrnei.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵)
7		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
8	1, 2, 4, 6, 7	ntrneiel 44070	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑋)))
9	8	notbid 318	. . 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 44066	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
12		elmapi 8887	. . . . . . 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 7104	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵)
15	14	elpwid 4613	. . . 4 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵)
16		biortn 937	. . . 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 3068	. . . 4 ⊢ (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁‘𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁‘𝑋)))
19		nss 4059	. . . 4 ⊢ (¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁‘𝑋)))
20	18, 19	bitr4i 278	. . 3 ⊢ (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁‘𝑋) ↔ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋))
21		df-ne 2938	. . . 4 ⊢ ((𝑁‘𝑋) ≠ 𝒫 𝐵 ↔ ¬ (𝑁‘𝑋) = 𝒫 𝐵)
22		ianor 983	. . . . 5 ⊢ (¬ ((𝑁‘𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁‘𝑋)) ↔ (¬ (𝑁‘𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋)))
23		eqss 4010	. . . . 5 ⊢ ((𝑁‘𝑋) = 𝒫 𝐵 ↔ ((𝑁‘𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁‘𝑋)))
24	22, 23	xchnxbir 333	. . . 4 ⊢ (¬ (𝑁‘𝑋) = 𝒫 𝐵 ↔ (¬ (𝑁‘𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋)))
25	21, 24	bitri 275	. . 3 ⊢ ((𝑁‘𝑋) ≠ 𝒫 𝐵 ↔ (¬ (𝑁‘𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋)))
26	17, 20, 25	3bitr4g 314	. 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁‘𝑋) ↔ (𝑁‘𝑋) ≠ 𝒫 𝐵))
27	10, 26	bitrd 279	1 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼‘𝑠) ↔ (𝑁‘𝑋) ≠ 𝒫 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1536  ∃wex 1775   ∈ wcel 2105   ≠ wne 2937  ∃wrex 3067  {crab 3432  Vcvv 3477   ⊆ wss 3962  𝒫 cpw 4604   class class class wbr 5147   ↦ cmpt 5230  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430   ∈ cmpo 7432   ↑_m cmap 8864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866

This theorem is referenced by:  ntrneineine1  44077