Theorem ntrneineine1lem 39870

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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
5		ntrnei.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6	5	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵)
7		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
8	1, 2, 4, 6, 7	ntrneiel 39867	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑋)))
9	8	notbid 319	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (¬ 𝑋 ∈ (𝐼‘𝑠) ↔ ¬ 𝑠 ∈ (𝑁‘𝑋)))
10	9	rexbidva 3256	. 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁‘𝑋)))
11	1, 2, 3	ntrneinex 39863	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑𝑚 𝐵))
12		elmapi 8269	. . . . . . 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	ffvelrnd 6708	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵)
15	14	elpwid 4459	. . . 4 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵)
16		biortn 930	. . . 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 3109	. . . 4 ⊢ (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁‘𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁‘𝑋)))
19		nss 3945	. . . 4 ⊢ (¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁‘𝑋)))
20	18, 19	bitr4i 279	. . 3 ⊢ (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁‘𝑋) ↔ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋))
21		df-ne 2983	. . . 4 ⊢ ((𝑁‘𝑋) ≠ 𝒫 𝐵 ↔ ¬ (𝑁‘𝑋) = 𝒫 𝐵)
22		ianor 974	. . . . 5 ⊢ (¬ ((𝑁‘𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁‘𝑋)) ↔ (¬ (𝑁‘𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋)))
23		eqss 3899	. . . . 5 ⊢ ((𝑁‘𝑋) = 𝒫 𝐵 ↔ ((𝑁‘𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁‘𝑋)))
24	22, 23	xchnxbir 334	. . . 4 ⊢ (¬ (𝑁‘𝑋) = 𝒫 𝐵 ↔ (¬ (𝑁‘𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋)))
25	21, 24	bitri 276	. . 3 ⊢ ((𝑁‘𝑋) ≠ 𝒫 𝐵 ↔ (¬ (𝑁‘𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋)))
26	17, 20, 25	3bitr4g 315	. 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁‘𝑋) ↔ (𝑁‘𝑋) ≠ 𝒫 𝐵))
27	10, 26	bitrd 280	1 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼‘𝑠) ↔ (𝑁‘𝑋) ≠ 𝒫 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   = wceq 1520  ∃wex 1759   ∈ wcel 2079   ≠ wne 2982  ∃wrex 3104  {crab 3107  Vcvv 3432   ⊆ wss 3854  𝒫 cpw 4447   class class class wbr 4956   ↦ cmpt 5035  ⟶wf 6213  ‘cfv 6217  (class class class)co 7007   ∈ cmpo 7009   ↑_𝑚 cmap 8247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-ov 7010  df-oprab 7011  df-mpo 7012  df-1st 7536  df-2nd 7537  df-map 8249

This theorem is referenced by:  ntrneineine1  39874