Theorem neicvgfv 41198

Description: The value of the neighborhoods (convergents) in terms of the the convergents (neighborhoods) function. (Contributed by RP, 27-Jun-2021.)

Hypotheses

Ref	Expression
neicvg.o	⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
neicvg.p	⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))
neicvg.d	⊢ 𝐷 = (𝑃‘𝐵)
neicvg.f	⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
neicvg.g	⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)
neicvg.h	⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))
neicvg.r	⊢ (𝜑 → 𝑁𝐻𝑀)
neicvgfv.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
neicvgfv	⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)})

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠   𝐵,𝑛,𝑜,𝑝,𝑠   𝐷,𝑖,𝑗,𝑘,𝑙,𝑚   𝐷,𝑛,𝑜,𝑝   𝑖,𝐹,𝑗,𝑘,𝑙   𝑛,𝐹,𝑜,𝑝   𝑖,𝐺,𝑗,𝑘,𝑙,𝑚   𝑛,𝐺,𝑜,𝑝   𝑖,𝑀,𝑗,𝑘,𝑙   𝑛,𝑀,𝑜,𝑝   𝑖,𝑁,𝑗,𝑘,𝑙,𝑚,𝑠   𝑛,𝑁,𝑜,𝑝   𝑋,𝑙,𝑚,𝑠   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠   𝜑,𝑛,𝑜,𝑝

Allowed substitution hints:   𝜑(𝑚)   𝐷(𝑠)   𝑃(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑠,𝑝,𝑙)   𝐹(𝑚,𝑠)   𝐺(𝑠)   𝐻(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑠,𝑝,𝑙)   𝑀(𝑚,𝑠)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑠,𝑝,𝑙)   𝑋(𝑖,𝑗,𝑘,𝑛,𝑜,𝑝)

Proof of Theorem neicvgfv

Step	Hyp	Ref	Expression
1		dfin5 3867	. 2 ⊢ (𝒫 𝐵 ∩ (𝑁‘𝑋)) = {𝑠 ∈ 𝒫 𝐵 ∣ 𝑠 ∈ (𝑁‘𝑋)}
2		neicvg.o	. . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
3		neicvg.p	. . . . . . 7 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))
4		neicvg.d	. . . . . . 7 ⊢ 𝐷 = (𝑃‘𝐵)
5		neicvg.f	. . . . . . 7 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
6		neicvg.g	. . . . . . 7 ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)
7		neicvg.h	. . . . . . 7 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))
8		neicvg.r	. . . . . . 7 ⊢ (𝜑 → 𝑁𝐻𝑀)
9	2, 3, 4, 5, 6, 7, 8	neicvgnex 41195	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
10		elmapi 8439	. . . . . 6 ⊢ (𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
12		neicvgfv.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
13	11, 12	ffvelrnd 6844	. . . 4 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵)
14	13	elpwid 4506	. . 3 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵)
15		sseqin2 4121	. . 3 ⊢ ((𝑁‘𝑋) ⊆ 𝒫 𝐵 ↔ (𝒫 𝐵 ∩ (𝑁‘𝑋)) = (𝑁‘𝑋))
16	14, 15	sylib 221	. 2 ⊢ (𝜑 → (𝒫 𝐵 ∩ (𝑁‘𝑋)) = (𝑁‘𝑋))
17	8	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑁𝐻𝑀)
18	12	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵)
19		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
20	2, 3, 4, 5, 6, 7, 17, 18, 19	neicvgel1 41196	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑠 ∈ (𝑁‘𝑋) ↔ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)))
21	20	rabbidva 3391	. 2 ⊢ (𝜑 → {𝑠 ∈ 𝒫 𝐵 ∣ 𝑠 ∈ (𝑁‘𝑋)} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)})
22	1, 16, 21	3eqtr3a 2818	1 ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  {crab 3075  Vcvv 3410   ∖ cdif 3856   ∩ cin 3858   ⊆ wss 3859  𝒫 cpw 4495   class class class wbr 5033   ↦ cmpt 5113   ∘ ccom 5529  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151   ∈ cmpo 7153   ↑_m cmap 8417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7694  df-2nd 7695  df-map 8419

This theorem is referenced by: (None)