filnet - Mathbox for Jeff Hankins

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Theorem filnet 32377

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 2622	. 2 ⊢ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)
2		eqid 2622	. 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 32376	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 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃wrex 2913 ⊆ wss 3574 {csn 4177 ∪ ciun 4520 {copab 4712 × cxp 5112 dom cdm 5114 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 DirRelcdir 17228 tailctail 17229 Filcfil 21649 FilMap cfm 21737

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-dir 17230 df-tail 17231 df-fbas 19743 df-fg 19744 df-fil 21650 df-fm 21742

This theorem is referenced by: (None)

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