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Theorem filnet 35315

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 2733	. 2 ⊢ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)
2		eqid 2733	. 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 35314	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃wrex 3071 ⊆ wss 3949 {csn 4629 ∪ ciun 4998 {copab 5211 × cxp 5675 dom cdm 5677 ran crn 5678 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 DirRelcdir 18547 tailctail 18548 Filcfil 23349 FilMap cfm 23437

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-dir 18549 df-tail 18550 df-fbas 20941 df-fg 20942 df-fil 23350 df-fm 23442

This theorem is referenced by: (None)

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