Theorem ntrneifv3 37400

Description: The value of the neighbors (convergents) expressed in terms of the interior (closure) 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)
ntrnei.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
ntrneifv3	⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑠)})

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠   𝑘,𝐼,𝑙,𝑚   𝑁,𝑠   𝑋,𝑙,𝑚,𝑠   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠

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

Proof of Theorem ntrneifv3

Step	Hyp	Ref	Expression
1		dfin5 3548	. 2 ⊢ (𝒫 𝐵 ∩ (𝑁‘𝑋)) = {𝑠 ∈ 𝒫 𝐵 ∣ 𝑠 ∈ (𝑁‘𝑋)}
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	ntrneinex 37395	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑𝑚 𝐵))
6		elmapi 7765	. . . . . 6 ⊢ (𝑁 ∈ (𝒫 𝒫 𝐵 ↑𝑚 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
8		ntrnei.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9	7, 8	ffvelrnd 6268	. . . 4 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵)
10	9	elpwid 4118	. . 3 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵)
11		sseqin2 3779	. . 3 ⊢ ((𝑁‘𝑋) ⊆ 𝒫 𝐵 ↔ (𝒫 𝐵 ∩ (𝑁‘𝑋)) = (𝑁‘𝑋))
12	10, 11	sylib 207	. 2 ⊢ (𝜑 → (𝒫 𝐵 ∩ (𝑁‘𝑋)) = (𝑁‘𝑋))
13	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
14	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵)
15		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
16	2, 3, 13, 14, 15	ntrneiel 37399	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑋)))
17	16	bicomd 212	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑠 ∈ (𝑁‘𝑋) ↔ 𝑋 ∈ (𝐼‘𝑠)))
18	17	rabbidva 3163	. 2 ⊢ (𝜑 → {𝑠 ∈ 𝒫 𝐵 ∣ 𝑠 ∈ (𝑁‘𝑋)} = {𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑠)})
19	1, 12, 18	3eqtr3a 2668	1 ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑠)})

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑_𝑚 cmap 7744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746

This theorem is referenced by: (None)