Theorem neicvgfv 44083

Description: The value of the neighborhoods (convergents) in terms of 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 3919	. 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 44080	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
10		elmapi 8799	. . . . . 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	ffvelcdmd 7039	. . . 4 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵)
14	13	elpwid 4568	. . 3 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵)
15		sseqin2 4182	. . 3 ⊢ ((𝑁‘𝑋) ⊆ 𝒫 𝐵 ↔ (𝒫 𝐵 ∩ (𝑁‘𝑋)) = (𝑁‘𝑋))
16	14, 15	sylib 218	. 2 ⊢ (𝜑 → (𝒫 𝐵 ∩ (𝑁‘𝑋)) = (𝑁‘𝑋))
17	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑁𝐻𝑀)
18	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵)
19		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
20	2, 3, 4, 5, 6, 7, 17, 18, 19	neicvgel1 44081	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑠 ∈ (𝑁‘𝑋) ↔ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)))
21	20	rabbidva 3409	. 2 ⊢ (𝜑 → {𝑠 ∈ 𝒫 𝐵 ∣ 𝑠 ∈ (𝑁‘𝑋)} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)})
22	1, 16, 21	3eqtr3a 2788	1 ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3402  Vcvv 3444   ∖ cdif 3908   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559   class class class wbr 5102   ↦ cmpt 5183   ∘ ccom 5635  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371   ↑_m cmap 8776

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778

This theorem is referenced by: (None)