Theorem ntrneiel2 41696

Description: Membership in iterated interior of a set is equivalent to there existing a particular neighborhood of that member such that points are members of that neighborhood if and only if the set is a neighborhood of each of those points. (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)
ntrneiel2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ntrneiel2.s	⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)

Assertion

Ref	Expression
ntrneiel2	⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐼‘𝑆)) ↔ ∃𝑢 ∈ (𝑁‘𝑋)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦))))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑦   𝑢,𝐵,𝑦   𝑘,𝐼,𝑙,𝑚,𝑦   𝑢,𝑁,𝑦   𝑆,𝑚,𝑦   𝑢,𝑆   𝑋,𝑙,𝑚,𝑦   𝑢,𝑋   𝜑,𝑖,𝑗,𝑘,𝑙,𝑦   𝜑,𝑢

Allowed substitution hints:   𝜑(𝑚)   𝑆(𝑖,𝑗,𝑘,𝑙)   𝐹(𝑦,𝑢,𝑖,𝑗,𝑘,𝑚,𝑙)   𝐼(𝑢,𝑖,𝑗)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑦,𝑢,𝑖,𝑗,𝑘,𝑚,𝑙)   𝑋(𝑖,𝑗,𝑘)

Proof of Theorem ntrneiel2

Step	Hyp	Ref	Expression
1		ntrnei.o	. . 3 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
2		ntrnei.f	. . 3 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
3		ntrnei.r	. . 3 ⊢ (𝜑 → 𝐼𝐹𝑁)
4		ntrneiel2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5	1, 2, 3	ntrneiiex 41686	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
6		elmapi 8637	. . . . 5 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
8		ntrneiel2.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
9	7, 8	ffvelrnd 6962	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) ∈ 𝒫 𝐵)
10	1, 2, 3, 4, 9	ntrneiel 41691	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐼‘𝑆)) ↔ (𝐼‘𝑆) ∈ (𝑁‘𝑋)))
11	1, 2, 3, 8	ntrneifv4 41695	. . . 4 ⊢ (𝜑 → (𝐼‘𝑆) = {𝑦 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑦)})
12		df-rab 3073	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑦)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))}
13	11, 12	eqtrdi 2794	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))})
14	13	eleq1d 2823	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) ∈ (𝑁‘𝑋) ↔ {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))} ∈ (𝑁‘𝑋)))
15		clabel 2885	. . . 4 ⊢ ({𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))} ∈ (𝑁‘𝑋) ↔ ∃𝑢(𝑢 ∈ (𝑁‘𝑋) ∧ ∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
16		df-rex 3070	. . . 4 ⊢ (∃𝑢 ∈ (𝑁‘𝑋)∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))) ↔ ∃𝑢(𝑢 ∈ (𝑁‘𝑋) ∧ ∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
17	15, 16	bitr4i 277	. . 3 ⊢ ({𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))} ∈ (𝑁‘𝑋) ↔ ∃𝑢 ∈ (𝑁‘𝑋)∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
18		ibar 529	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (𝑆 ∈ (𝑁‘𝑦) ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
19	18	bibi2d 343	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → ((𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦)) ↔ (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
20	19	ralbiia 3091	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
21		ssv 3945	. . . . . . . 8 ⊢ 𝐵 ⊆ V
22	21	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → 𝐵 ⊆ V)
23		vex 3436	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
24		eldif 3897	. . . . . . . . . 10 ⊢ (𝑦 ∈ (V ∖ 𝐵) ↔ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ 𝐵))
25	23, 24	mpbiran 706	. . . . . . . . 9 ⊢ (𝑦 ∈ (V ∖ 𝐵) ↔ ¬ 𝑦 ∈ 𝐵)
26	1, 2, 3	ntrneinex 41687	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
27		elmapi 8637	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
28	26, 27	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
29	28, 4	ffvelrnd 6962	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵)
30	29	elpwid 4544	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵)
31	30	sselda 3921	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → 𝑢 ∈ 𝒫 𝐵)
32	31	elpwid 4544	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → 𝑢 ⊆ 𝐵)
33	32	sseld 3920	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (𝑦 ∈ 𝑢 → 𝑦 ∈ 𝐵))
34	33	con3dimp 409	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) ∧ ¬ 𝑦 ∈ 𝐵) → ¬ 𝑦 ∈ 𝑢)
35		pm3.14 993	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑦 ∈ 𝐵 ∨ ¬ 𝑆 ∈ (𝑁‘𝑦)) → ¬ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))
36	35	orcs 872	. . . . . . . . . . . 12 ⊢ (¬ 𝑦 ∈ 𝐵 → ¬ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))
37	36	adantl 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) ∧ ¬ 𝑦 ∈ 𝐵) → ¬ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))
38	34, 37	2falsed 377	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) ∧ ¬ 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
39	38	ex 413	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (¬ 𝑦 ∈ 𝐵 → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
40	25, 39	syl5bi 241	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (𝑦 ∈ (V ∖ 𝐵) → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
41	40	ralrimiv 3102	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → ∀𝑦 ∈ (V ∖ 𝐵)(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
42	22, 41	raldifeq 4424	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))) ↔ ∀𝑦 ∈ V (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
43	20, 42	bitrid 282	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦)) ↔ ∀𝑦 ∈ V (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
44		ralv 3456	. . . . 5 ⊢ (∀𝑦 ∈ V (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))) ↔ ∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
45	43, 44	bitr2di 288	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))) ↔ ∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦))))
46	45	rexbidva 3225	. . 3 ⊢ (𝜑 → (∃𝑢 ∈ (𝑁‘𝑋)∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))) ↔ ∃𝑢 ∈ (𝑁‘𝑋)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦))))
47	17, 46	bitrid 282	. 2 ⊢ (𝜑 → ({𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))} ∈ (𝑁‘𝑋) ↔ ∃𝑢 ∈ (𝑁‘𝑋)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦))))
48	10, 14, 47	3bitrd 305	1 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐼‘𝑆)) ↔ ∃𝑢 ∈ (𝑁‘𝑋)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  𝒫 cpw 4533   class class class wbr 5074   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   ↑_m cmap 8615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617

This theorem is referenced by:  ntrneik4  41711