Theorem ntrneiel2 44739

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 44729	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
6		elmapi 8846	. . . . 5 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
7	5, 6	syl 18	. . . 4 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
8		ntrneiel2.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
9	7, 8	ffvelcdmd 7081	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) ∈ 𝒫 𝐵)
10	1, 2, 3, 4, 9	ntrneiel 44734	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐼‘𝑆)) ↔ (𝐼‘𝑆) ∈ (𝑁‘𝑋)))
11	1, 2, 3, 8	ntrneifv4 44738	. . . 4 ⊢ (𝜑 → (𝐼‘𝑆) = {𝑦 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑦)})
12		df-rab 3423	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑦)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))}
13	11, 12	eqtrdi 2820	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))})
14	13	eleq1d 2854	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) ∈ (𝑁‘𝑋) ↔ {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))} ∈ (𝑁‘𝑋)))
15		clabel 2914	. . . 4 ⊢ ({𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))} ∈ (𝑁‘𝑋) ↔ ∃𝑢(𝑢 ∈ (𝑁‘𝑋) ∧ ∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
16		df-rex 3096	. . . 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 281	. . 3 ⊢ ({𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))} ∈ (𝑁‘𝑋) ↔ ∃𝑢 ∈ (𝑁‘𝑋)∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
18		ibar 537	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (𝑆 ∈ (𝑁‘𝑦) ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
19	18	bibi2d 345	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → ((𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦)) ↔ (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
20	19	ralbiia 3115	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
21		ssv 3967	. . . . . . . 8 ⊢ 𝐵 ⊆ V
22	21	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → 𝐵 ⊆ V)
23		vex 3465	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
24		eldif 3921	. . . . . . . . . 10 ⊢ (𝑦 ∈ (V ∖ 𝐵) ↔ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ 𝐵))
25	23, 24	mpbiran 721	. . . . . . . . 9 ⊢ (𝑦 ∈ (V ∖ 𝐵) ↔ ¬ 𝑦 ∈ 𝐵)
26	1, 2, 3	ntrneinex 44730	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
27		elmapi 8846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
28	26, 27	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
29	28, 4	ffvelcdmd 7081	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵)
30	29	elpwid 4574	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵)
31	30	sselda 3943	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → 𝑢 ∈ 𝒫 𝐵)
32	31	elpwid 4574	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → 𝑢 ⊆ 𝐵)
33	32	sseld 3942	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (𝑦 ∈ 𝑢 → 𝑦 ∈ 𝐵))
34	33	con3dimp 413	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) ∧ ¬ 𝑦 ∈ 𝐵) → ¬ 𝑦 ∈ 𝑢)
35		pm3.14 1011	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑦 ∈ 𝐵 ∨ ¬ 𝑆 ∈ (𝑁‘𝑦)) → ¬ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))
36	35	orcs 888	. . . . . . . . . . . 12 ⊢ (¬ 𝑦 ∈ 𝐵 → ¬ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))
37	36	adantl 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) ∧ ¬ 𝑦 ∈ 𝐵) → ¬ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))
38	34, 37	2falsed 379	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) ∧ ¬ 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
39	38	ex 417	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (¬ 𝑦 ∈ 𝐵 → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
40	25, 39	biimtrid 245	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (𝑦 ∈ (V ∖ 𝐵) → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
41	40	ralrimiv 3162	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → ∀𝑦 ∈ (V ∖ 𝐵)(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
42	22, 41	raldifeq 4457	. . . . . 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 286	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦)) ↔ ∀𝑦 ∈ V (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
44		ralv 3487	. . . . 5 ⊢ (∀𝑦 ∈ V (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))) ↔ ∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
45	43, 44	bitr2di 291	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))) ↔ ∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦))))
46	45	rexbidva 3193	. . 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 286	. 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 308	1 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐼‘𝑆)) ↔ ∃𝑢 ∈ (𝑁‘𝑋)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∃wrex 3095  {crab 3422  Vcvv 3461   ∖ cdif 3908   ⊆ wss 3911  𝒫 cpw 4565   class class class wbr 5111   ↦ cmpt 5194  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413   ↑_m cmap 8824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-map 8826

This theorem is referenced by:  ntrneik4  44754