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Theorem filnet 33346

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 2795	. 2 ⊢ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)
2		eqid 2795	. 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 33345	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 396 = wceq 1522 ∃wex 1761 ∈ wcel 2081 ∃wrex 3106 ⊆ wss 3863 {csn 4476 ∪ ciun 4829 {copab 5028 × cxp 5446 dom cdm 5448 ran crn 5449 ⟶wf 6226 ‘cfv 6230 (class class class)co 7021 1^st c1st 7548 DirRelcdir 17672 tailctail 17673 Filcfil 22142 FilMap cfm 22230

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-op 4483 df-uni 4750 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-id 5353 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-ov 7024 df-oprab 7025 df-mpo 7026 df-1st 7550 df-2nd 7551 df-dir 17674 df-tail 17675 df-fbas 20229 df-fg 20230 df-fil 22143 df-fm 22235

This theorem is referenced by: (None)

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