Theorem ntrneicls00 45281

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 45268	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
5		elmapi 8897	. . . . . 6 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
6	4, 5	syl 18	. . . . 5 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
7	1, 2, 3	ntrneibex 45265	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
8		pwidg 4581	. . . . . 6 ⊢ (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
9	7, 8	syl 18	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵)
10	6, 9	ffvelcdmd 7109	. . . 4 ⊢ (𝜑 → (𝐼‘𝐵) ∈ 𝒫 𝐵)
11	10	elpwid 4569	. . 3 ⊢ (𝜑 → (𝐼‘𝐵) ⊆ 𝐵)
12		eqss 3928	. . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 ↔ ((𝐼‘𝐵) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐼‘𝐵)))
13		dfss3 3902	. . . . . 6 ⊢ (𝐵 ⊆ (𝐼‘𝐵) ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵))
14	13	anbi2i 635	. . . . 5 ⊢ (((𝐼‘𝐵) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐼‘𝐵)) ↔ ((𝐼‘𝐵) ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵)))
15	12, 14	bitri 278	. . . 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 720	. 2 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵)))
18	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
19		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
20	9	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ 𝒫 𝐵)
21	1, 2, 18, 19, 20	ntrneiel 45273	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝐵) ↔ 𝐵 ∈ (𝑁‘𝑥)))
22	21	ralbidva 3187	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵) ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))
23	17, 22	bitrd 282	1 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560   class class class wbr 5106   ↦ cmpt 5189  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445   ↑_m cmap 8875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-map 8877

This theorem is used by: (None)