ntrneik4w - Mathbox for Richard Penner

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Theorem ntrneik4w 43530

Description: Idempotence of the interior function is equivalent to saying a set is a neighborhood of a point if and only if the interior of the set is a neighborhood of a point. (Contributed by RP, 11-Jul-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
ntrneik4w	⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥))))

Distinct variable groups: 𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠,𝑥 𝑘,𝐼,𝑙,𝑚,𝑥 𝜑,𝑖,𝑗,𝑘,𝑙,𝑠,𝑥 Allowed substitution hints: 𝜑(𝑚) 𝐹(𝑥,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙) 𝐼(𝑖,𝑗,𝑠) 𝑁(𝑥,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙) 𝑂(𝑥,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)

**Proof of Theorem ntrneik4w**
Step	Hyp	Ref	Expression
1		dfcleq 2721	. . . . 5 ⊢ ((𝐼‘𝑠) = (𝐼‘(𝐼‘𝑠)) ↔ ∀𝑥(𝑥 ∈ (𝐼‘𝑠) ↔ 𝑥 ∈ (𝐼‘(𝐼‘𝑠))))
2		eqcom 2735	. . . . 5 ⊢ ((𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ (𝐼‘𝑠) = (𝐼‘(𝐼‘𝑠)))
3		ralv 3496	. . . . 5 ⊢ (∀𝑥 ∈ V (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑥 ∈ (𝐼‘(𝐼‘𝑠))) ↔ ∀𝑥(𝑥 ∈ (𝐼‘𝑠) ↔ 𝑥 ∈ (𝐼‘(𝐼‘𝑠))))
4	1, 2, 3	3bitr4i 303	. . . 4 ⊢ ((𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ V (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑥 ∈ (𝐼‘(𝐼‘𝑠))))
5		ssv 4004	. . . . . . 7 ⊢ 𝐵 ⊆ V
6	5	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐵 ⊆ V)
7		vex 3475	. . . . . . . . 9 ⊢ 𝑥 ∈ V
8		eldif 3957	. . . . . . . . 9 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐵))
9	7, 8	mpbiran 708	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥 ∈ 𝐵)
10		ntrnei.o	. . . . . . . . . . . . . . . 16 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
11		ntrnei.f	. . . . . . . . . . . . . . . 16 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
12		ntrnei.r	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼𝐹𝑁)
13	10, 11, 12	ntrneiiex 43506	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))
14		elmapi 8868	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
15	13, 14	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
16	15	ffvelcdmda 7094	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ∈ 𝒫 𝐵)
17	16	elpwid 4612	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ⊆ 𝐵)
18	17	sseld 3979	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑥 ∈ (𝐼‘𝑠) → 𝑥 ∈ 𝐵))
19	18	con3dimp 408	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ ¬ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ (𝐼‘𝑠))
20	15	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
21	20, 16	ffvelcdmd 7095	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘(𝐼‘𝑠)) ∈ 𝒫 𝐵)
22	21	elpwid 4612	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘(𝐼‘𝑠)) ⊆ 𝐵)
23	22	sseld 3979	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑥 ∈ (𝐼‘(𝐼‘𝑠)) → 𝑥 ∈ 𝐵))
24	23	con3dimp 408	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ ¬ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ (𝐼‘(𝐼‘𝑠)))
25	19, 24	2falsed 376	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ ¬ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑥 ∈ (𝐼‘(𝐼‘𝑠))))
26	25	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (¬ 𝑥 ∈ 𝐵 → (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑥 ∈ (𝐼‘(𝐼‘𝑠)))))
27	9, 26	biimtrid 241	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑥 ∈ (V ∖ 𝐵) → (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑥 ∈ (𝐼‘(𝐼‘𝑠)))))
28	27	ralrimiv 3142	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → ∀𝑥 ∈ (V ∖ 𝐵)(𝑥 ∈ (𝐼‘𝑠) ↔ 𝑥 ∈ (𝐼‘(𝐼‘𝑠))))
29	6, 28	raldifeq 4494	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑥 ∈ (𝐼‘(𝐼‘𝑠))) ↔ ∀𝑥 ∈ V (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑥 ∈ (𝐼‘(𝐼‘𝑠)))))
30	12	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
31	30	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
32		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
33		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑠 ∈ 𝒫 𝐵)
34	10, 11, 31, 32, 33	ntrneiel 43511	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑥)))
35	16	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑠) ∈ 𝒫 𝐵)
36	10, 11, 31, 32, 35	ntrneiel 43511	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘(𝐼‘𝑠)) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥)))
37	34, 36	bibi12d 345	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 ∈ (𝐼‘𝑠) ↔ 𝑥 ∈ (𝐼‘(𝐼‘𝑠))) ↔ (𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥))))
38	37	ralbidva 3172	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑥 ∈ (𝐼‘(𝐼‘𝑠))) ↔ ∀𝑥 ∈ 𝐵 (𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥))))
39	29, 38	bitr3d 281	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (∀𝑥 ∈ V (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑥 ∈ (𝐼‘(𝐼‘𝑠))) ↔ ∀𝑥 ∈ 𝐵 (𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥))))
40	4, 39	bitrid 283	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 (𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥))))
41	40	ralbidva 3172	. 2 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑥 ∈ 𝐵 (𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥))))
42		ralcom 3283	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑥 ∈ 𝐵 (𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥)))
43	41, 42	bitrdi 287	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1532 = wceq 1534 ∈ wcel 2099 ∀wral 3058 {crab 3429 Vcvv 3471 ∖ cdif 3944 ⊆ wss 3947 𝒫 cpw 4603 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 ↑_m cmap 8845

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-map 8847

This theorem is referenced by: ntrneik4 43531

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