Theorem ntrneineine1lem 44469

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 44466	. . . 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 3160	. 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁‘𝑋)))
11	1, 2, 3	ntrneinex 44462	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
12		elmapi 8800	. . . . . . 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 7041	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵)
15	14	elpwid 4565	. . . 4 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵)
16		biortn 938	. . . 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 3063	. . . 4 ⊢ (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁‘𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁‘𝑋)))
19		nss 4000	. . . 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 2934	. . . 4 ⊢ ((𝑁‘𝑋) ≠ 𝒫 𝐵 ↔ ¬ (𝑁‘𝑋) = 𝒫 𝐵)
22		ianor 984	. . . . 5 ⊢ (¬ ((𝑁‘𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁‘𝑋)) ↔ (¬ (𝑁‘𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁‘𝑋)))
23		eqss 3951	. . . . 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 848   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  {crab 3401  Vcvv 3442   ⊆ wss 3903  𝒫 cpw 4556   class class class wbr 5100   ↦ cmpt 5181  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370   ∈ cmpo 7372   ↑_m cmap 8777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946  df-map 8779

This theorem is referenced by:  ntrneineine1  44473