Theorem ntrneicls00 44051

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 44038	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
5		elmapi 8907	. . . . . 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 44035	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
8		pwidg 4642	. . . . . 6 ⊢ (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵)
10	6, 9	ffvelcdmd 7119	. . . 4 ⊢ (𝜑 → (𝐼‘𝐵) ∈ 𝒫 𝐵)
11	10	elpwid 4631	. . 3 ⊢ (𝜑 → (𝐼‘𝐵) ⊆ 𝐵)
12		eqss 4024	. . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 ↔ ((𝐼‘𝐵) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐼‘𝐵)))
13		dfss3 3997	. . . . . 6 ⊢ (𝐵 ⊆ (𝐼‘𝐵) ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵))
14	13	anbi2i 622	. . . . 5 ⊢ (((𝐼‘𝐵) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐼‘𝐵)) ↔ ((𝐼‘𝐵) ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵)))
15	12, 14	bitri 275	. . . 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 706	. 2 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵)))
18	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
19		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
20	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ 𝒫 𝐵)
21	1, 2, 18, 19, 20	ntrneiel 44043	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝐵) ↔ 𝐵 ∈ (𝑁‘𝑥)))
22	21	ralbidva 3182	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵) ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))
23	17, 22	bitrd 279	1 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622   class class class wbr 5166   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450   ↑_m cmap 8884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-map 8886

This theorem is referenced by: (None)