Theorem ntrneifv4 44062

Description: The value of the interior (closure) expressed in terms of the neighbors (convergents) function. (Contributed by RP, 26-Jun-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)
ntrneifv.s	⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)

Assertion

Ref	Expression
ntrneifv4	⊢ (𝜑 → (𝐼‘𝑆) = {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)})

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑥   𝑘,𝐼,𝑙,𝑚,𝑥   𝑆,𝑚,𝑥   𝜑,𝑖,𝑗,𝑘,𝑙,𝑥

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

Proof of Theorem ntrneifv4

Step	Hyp	Ref	Expression
1		dfin5 3911	. 2 ⊢ (𝐵 ∩ (𝐼‘𝑆)) = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ (𝐼‘𝑆)}
2		ntrnei.o	. . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
3		ntrnei.f	. . . . . . 7 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
4		ntrnei.r	. . . . . . 7 ⊢ (𝜑 → 𝐼𝐹𝑁)
5	2, 3, 4	ntrneiiex 44053	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
6		elmapi 8776	. . . . . 6 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
8		ntrneifv.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
9	7, 8	ffvelcdmd 7019	. . . 4 ⊢ (𝜑 → (𝐼‘𝑆) ∈ 𝒫 𝐵)
10	9	elpwid 4560	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) ⊆ 𝐵)
11		sseqin2 4174	. . 3 ⊢ ((𝐼‘𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐼‘𝑆)) = (𝐼‘𝑆))
12	10, 11	sylib 218	. 2 ⊢ (𝜑 → (𝐵 ∩ (𝐼‘𝑆)) = (𝐼‘𝑆))
13	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
14		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
15	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑆 ∈ 𝒫 𝐵)
16	2, 3, 13, 14, 15	ntrneiel 44058	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑆) ↔ 𝑆 ∈ (𝑁‘𝑥)))
17	16	rabbidva 3401	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ (𝐼‘𝑆)} = {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)})
18	1, 12, 17	3eqtr3a 2788	1 ⊢ (𝜑 → (𝐼‘𝑆) = {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3394  Vcvv 3436   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4551   class class class wbr 5092   ↦ cmpt 5173  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351   ↑_m cmap 8753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-map 8755

This theorem is referenced by:  ntrneiel2  44063