Theorem ntrneiel2 39082

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 39072	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
6		elmapi 8086	. . . . 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 6554	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) ∈ 𝒫 𝐵)
10	1, 2, 3, 4, 9	ntrneiel 39077	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐼‘𝑆)) ↔ (𝐼‘𝑆) ∈ (𝑁‘𝑋)))
11	1, 2, 3, 8	ntrneifv4 39081	. . . 4 ⊢ (𝜑 → (𝐼‘𝑆) = {𝑦 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑦)})
12		df-rab 3064	. . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑦)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))}
13	11, 12	syl6eq 2815	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))})
14	13	eleq1d 2829	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) ∈ (𝑁‘𝑋) ↔ {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))} ∈ (𝑁‘𝑋)))
15		clabel 2892	. . . 4 ⊢ ({𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))} ∈ (𝑁‘𝑋) ↔ ∃𝑢(𝑢 ∈ (𝑁‘𝑋) ∧ ∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
16		df-rex 3061	. . . 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 269	. . 3 ⊢ ({𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))} ∈ (𝑁‘𝑋) ↔ ∃𝑢 ∈ (𝑁‘𝑋)∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
18		ibar 524	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (𝑆 ∈ (𝑁‘𝑦) ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
19	18	bibi2d 333	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → ((𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦)) ↔ (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
20	19	ralbiia 3126	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
21		ssv 3787	. . . . . . . 8 ⊢ 𝐵 ⊆ V
22	21	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → 𝐵 ⊆ V)
23		vex 3353	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
24		eldif 3744	. . . . . . . . . 10 ⊢ (𝑦 ∈ (V ∖ 𝐵) ↔ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ 𝐵))
25	23, 24	mpbiran 700	. . . . . . . . 9 ⊢ (𝑦 ∈ (V ∖ 𝐵) ↔ ¬ 𝑦 ∈ 𝐵)
26	1, 2, 3	ntrneinex 39073	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑𝑚 𝐵))
27		elmapi 8086	. . . . . . . . . . . . . . . . . 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 6554	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵)
30	29	elpwid 4329	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵)
31	30	sselda 3763	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → 𝑢 ∈ 𝒫 𝐵)
32	31	elpwid 4329	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → 𝑢 ⊆ 𝐵)
33	32	sseld 3762	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (𝑦 ∈ 𝑢 → 𝑦 ∈ 𝐵))
34	33	con3dimp 397	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) ∧ ¬ 𝑦 ∈ 𝐵) → ¬ 𝑦 ∈ 𝑢)
35		pm3.14 1018	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑦 ∈ 𝐵 ∨ ¬ 𝑆 ∈ (𝑁‘𝑦)) → ¬ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))
36	35	orcs 901	. . . . . . . . . . . 12 ⊢ (¬ 𝑦 ∈ 𝐵 → ¬ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))
37	36	adantl 473	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) ∧ ¬ 𝑦 ∈ 𝐵) → ¬ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))
38	34, 37	2falsed 367	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) ∧ ¬ 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
39	38	ex 401	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (¬ 𝑦 ∈ 𝐵 → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
40	25, 39	syl5bi 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (𝑦 ∈ (V ∖ 𝐵) → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
41	40	ralrimiv 3112	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → ∀𝑦 ∈ (V ∖ 𝐵)(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
42	22, 41	raldifeq 4220	. . . . . 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	syl5bb 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦)) ↔ ∀𝑦 ∈ V (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦)))))
44		ralv 3372	. . . . 5 ⊢ (∀𝑦 ∈ V (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))) ↔ ∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))))
45	43, 44	syl6rbb 279	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑁‘𝑋)) → (∀𝑦(𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝐵 ∧ 𝑆 ∈ (𝑁‘𝑦))) ↔ ∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦))))
46	45	rexbidva 3196	. . 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	syl5bb 274	. 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 296	1 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐼‘𝑆)) ↔ ∃𝑢 ∈ (𝑁‘𝑋)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384  ∀wal 1650   = wceq 1652  ∃wex 1874   ∈ wcel 2155  {cab 2751  ∀wral 3055  ∃wrex 3056  {crab 3059  Vcvv 3350   ∖ cdif 3731   ⊆ wss 3734  𝒫 cpw 4317   class class class wbr 4811   ↦ cmpt 4890  ⟶wf 6066  ‘cfv 6070  (class class class)co 6846   ↦ cmpt2 6848   ↑_𝑚 cmap 8064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-1st 7370  df-2nd 7371  df-map 8066

This theorem is referenced by:  ntrneik4  39097