filnet - Mathbox for Jeff Hankins

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Theorem filnet 37004

Description: A filter has the same convergence and clustering properties as some net. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

**Assertion**
Ref	Expression
filnet	⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))

Distinct variable groups: 𝑓,𝑑,𝐹 𝑋,𝑑,𝑓

Proof of Theorem filnet Dummy variables 𝑥 𝑦 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. 2 ⊢ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)
2		eqid 2760	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ∧ 𝑦 ∈ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)) ∧ (1^st ‘𝑦) ⊆ (1^st ‘𝑥))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ∧ 𝑦 ∈ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)) ∧ (1^st ‘𝑦) ⊆ (1^st ‘𝑥))}
3	1, 2	filnetlem4 37003	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2145 ∃wrex 3086 ⊆ wss 3899 {csn 4584 ∪ ciun 4951 {copab 5167 × cxp 5653 dom cdm 5655 ran crn 5656 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 1^st c1st 7985 DirRelcdir 18685 tailctail 18686 Filcfil 24074 FilMap cfm 24162

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-dir 18687 df-tail 18688 df-fbas 21585 df-fg 21586 df-fil 24075 df-fm 24167

This theorem is used by: (None)

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