Theorem ntrneicls00 38907

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 2-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)

Assertion

Ref	Expression
ntrneicls00	⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑥   𝑘,𝐼,𝑙,𝑚,𝑥   𝜑,𝑖,𝑗,𝑘,𝑙,𝑥

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

Proof of Theorem ntrneicls00

Step	Hyp	Ref	Expression
1		ntrnei.o	. . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
2		ntrnei.f	. . . . . . 7 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
3		ntrnei.r	. . . . . . 7 ⊢ (𝜑 → 𝐼𝐹𝑁)
4	1, 2, 3	ntrneiiex 38894	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
5		elmapi 8047	. . . . . 6 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
7	1, 2, 3	ntrneibex 38891	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
8		pwidg 4317	. . . . . 6 ⊢ (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵)
10	6, 9	ffvelrnd 6524	. . . 4 ⊢ (𝜑 → (𝐼‘𝐵) ∈ 𝒫 𝐵)
11	10	elpwid 4314	. . 3 ⊢ (𝜑 → (𝐼‘𝐵) ⊆ 𝐵)
12		eqss 3759	. . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 ↔ ((𝐼‘𝐵) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐼‘𝐵)))
13		dfss3 3733	. . . . . 6 ⊢ (𝐵 ⊆ (𝐼‘𝐵) ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵))
14	13	anbi2i 732	. . . . 5 ⊢ (((𝐼‘𝐵) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐼‘𝐵)) ↔ ((𝐼‘𝐵) ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵)))
15	12, 14	bitri 264	. . . 4 ⊢ ((𝐼‘𝐵) = 𝐵 ↔ ((𝐼‘𝐵) ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵)))
16	15	a1i 11	. . 3 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ((𝐼‘𝐵) ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵))))
17	11, 16	mpbirand 531	. 2 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵)))
18	3	adantr 472	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
19		simpr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
20	9	adantr 472	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ 𝒫 𝐵)
21	1, 2, 18, 19, 20	ntrneiel 38899	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝐵) ↔ 𝐵 ∈ (𝑁‘𝑥)))
22	21	ralbidva 3123	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵) ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))
23	17, 22	bitrd 268	1 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  {crab 3054  Vcvv 3340   ⊆ wss 3715  𝒫 cpw 4302   class class class wbr 4804   ↦ cmpt 4881  ⟶wf 6045  ‘cfv 6049  (class class class)co 6814   ↦ cmpt2 6816   ↑_𝑚 cmap 8025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027

This theorem is referenced by: (None)