Theorem ntrneiel2 43658

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 43648	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
6		elmapi 8868	. . . . 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	ffvelcdmd 7094	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) ∈ 𝒫 𝐵)
10	1, 2, 3, 4, 9	ntrneiel 43653	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐼‘𝑆)) ↔ (𝐼‘𝑆) ∈ (𝑁‘𝑋)))
11	1, 2, 3, 8	ntrneifv4 43657	. . . 4 ⊢ (𝜑 → (𝐼‘𝑆) = {𝑦 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑦)})
12		df-rab 3419	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑦)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))}
13	11, 12	eqtrdi 2781	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))})
14	13	eleq1d 2810	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) ∈ (𝑁‘𝑋) ↔ {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))} ∈ (𝑁‘𝑋)))
15		clabel 2873	. . . 4 ⊢ ({𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))} ∈ (𝑁‘𝑋) ↔ ∃𝑢(𝑢 ∈ (𝑁‘𝑋) ∧ ∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
16		df-rex 3060	. . . 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 527	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (𝑆 ∈ (𝑁‘𝑦) ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
19	18	bibi2d 341	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → ((𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦)) ↔ (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
20	19	ralbiia 3080	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
21		ssv 4001	. . . . . . . 8 ⊢ 𝐵 ⊆ V
22	21	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → 𝐵 ⊆ V)
23		vex 3465	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
24		eldif 3954	. . . . . . . . . 10 ⊢ (𝑦 ∈ (V ∖ 𝐵) ↔ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ 𝐵))
25	23, 24	mpbiran 707	. . . . . . . . 9 ⊢ (𝑦 ∈ (V ∖ 𝐵) ↔ ¬ 𝑦 ∈ 𝐵)
26	1, 2, 3	ntrneinex 43649	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
27		elmapi 8868	. . . . . . . . . . . . . . . . . 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	ffvelcdmd 7094	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵)
30	29	elpwid 4613	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵)
31	30	sselda 3976	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → 𝑢 ∈ 𝒫 𝐵)
32	31	elpwid 4613	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → 𝑢 ⊆ 𝐵)
33	32	sseld 3975	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (𝑦 ∈ 𝑢 → 𝑦 ∈ 𝐵))
34	33	con3dimp 407	. . . . . . . . . . 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 873	. . . . . . . . . . . 12 ⊢ (¬ 𝑦 ∈ 𝐵 → ¬ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))
37	36	adantl 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) ∧ ¬ 𝑦 ∈ 𝐵) → ¬ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))
38	34, 37	2falsed 375	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) ∧ ¬ 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
39	38	ex 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (¬ 𝑦 ∈ 𝐵 → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
40	25, 39	biimtrid 241	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (𝑦 ∈ (V ∖ 𝐵) → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
41	40	ralrimiv 3134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → ∀𝑦 ∈ (V ∖ 𝐵)(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
42	22, 41	raldifeq 4495	. . . . . 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 3487	. . . . 5 ⊢ (∀𝑦 ∈ V (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))) ↔ ∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
45	43, 44	bitr2di 287	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))) ↔ ∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦))))
46	45	rexbidva 3166	. . 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 304	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 394  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2702  ∀wral 3050  ∃wrex 3059  {crab 3418  Vcvv 3461   ∖ cdif 3941   ⊆ wss 3944  𝒫 cpw 4604   class class class wbr 5149   ↦ cmpt 5232  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419   ∈ cmpo 7421   ↑_m cmap 8845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  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 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847

This theorem is referenced by:  ntrneik4  43673