Theorem neicvgel2 41619

Description: The complement of a subset being an element of a neighborhood at a point is equivalent to that subset not being a element of the convergent at that point. (Contributed by RP, 12-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)
neicvgel.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
neicvgel.s	⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)

Assertion

Ref	Expression
neicvgel2	⊢ (𝜑 → ((𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋) ↔ ¬ 𝑆 ∈ (𝑀‘𝑋)))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚   𝐵,𝑛,𝑜,𝑝   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   𝑛,𝑁,𝑜,𝑝   𝑆,𝑚   𝑆,𝑜   𝑋,𝑙,𝑚   𝜑,𝑖,𝑗,𝑘,𝑙   𝜑,𝑛,𝑜,𝑝

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

Proof of Theorem neicvgel2

Step	Hyp	Ref	Expression
1		neicvg.o	. . 3 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
2		neicvg.p	. . 3 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))
3		neicvg.d	. . 3 ⊢ 𝐷 = (𝑃‘𝐵)
4		neicvg.f	. . 3 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
5		neicvg.g	. . 3 ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)
6		neicvg.h	. . 3 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))
7		neicvg.r	. . 3 ⊢ (𝜑 → 𝑁𝐻𝑀)
8		neicvgel.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9	3, 6, 7	neicvgrcomplex 41612	. . 3 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	neicvgel1 41618	. 2 ⊢ (𝜑 → ((𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋) ↔ ¬ (𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑀‘𝑋)))
11		neicvgel.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
12	11	elpwid 4541	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
13		dfss4 4189	. . . . 5 ⊢ (𝑆 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑆)) = 𝑆)
14	12, 13	sylib 217	. . . 4 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ 𝑆)) = 𝑆)
15	14	eleq1d 2823	. . 3 ⊢ (𝜑 → ((𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑀‘𝑋) ↔ 𝑆 ∈ (𝑀‘𝑋)))
16	15	notbid 317	. 2 ⊢ (𝜑 → (¬ (𝐵 ∖ (𝐵 ∖ 𝑆)) ∈ (𝑀‘𝑋) ↔ ¬ 𝑆 ∈ (𝑀‘𝑋)))
17	10, 16	bitrd 278	1 ⊢ (𝜑 → ((𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋) ↔ ¬ 𝑆 ∈ (𝑀‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  𝒫 cpw 4530   class class class wbr 5070   ↦ cmpt 5153   ∘ ccom 5584  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257   ↑_m cmap 8573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575

This theorem is referenced by: (None)