Theorem ntrneicls00 44613

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 44600	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
5		elmapi 8819	. . . . . 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 44597	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
8		pwidg 4569	. . . . . 6 ⊢ (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵)
10	6, 9	ffvelcdmd 7055	. . . 4 ⊢ (𝜑 → (𝐼‘𝐵) ∈ 𝒫 𝐵)
11	10	elpwid 4558	. . 3 ⊢ (𝜑 → (𝐼‘𝐵) ⊆ 𝐵)
12		eqss 3946	. . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 ↔ ((𝐼‘𝐵) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐼‘𝐵)))
13		dfss3 3920	. . . . . 6 ⊢ (𝐵 ⊆ (𝐼‘𝐵) ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵))
14	13	anbi2i 631	. . . . 5 ⊢ (((𝐼‘𝐵) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐼‘𝐵)) ↔ ((𝐼‘𝐵) ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵)))
15	12, 14	bitri 277	. . . 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 715	. 2 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵)))
18	3	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
19		simpr 487	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
20	9	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ 𝒫 𝐵)
21	1, 2, 18, 19, 20	ntrneiel 44605	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝐵) ↔ 𝐵 ∈ (𝑁‘𝑥)))
22	21	ralbidva 3177	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝑥 ∈ (𝐼‘𝐵) ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))
23	17, 22	bitrd 281	1 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1554   ∈ wcel 2136  ∀wral 3070  {crab 3408  Vcvv 3448   ⊆ wss 3899  𝒫 cpw 4549   class class class wbr 5094   ↦ cmpt 5175  ⟶wf 6506  ‘cfv 6510  (class class class)co 7385   ∈ cmpo 7387   ↑_m cmap 8796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-1st 7959  df-2nd 7960  df-map 8798

This theorem is referenced by: (None)