filnet - Mathbox for Jeff Hankins

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Theorem filnet 37297

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 2763	. 2 ⊢ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)
2		eqid 2763	. 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 37296	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 1569 ∃wex 1811 ∈ wcel 2146 ∃wrex 3089 ⊆ wss 3899 {csn 4587 ∪ ciun 4953 {copab 5169 × cxp 5652 dom cdm 5654 ran crn 5655 ⟶wf 6542 ‘cfv 6546 (class class class)co 7427 1^st c1st 8006 DirRelcdir 18780 tailctail 18781 Filcfil 24176 FilMap cfm 24264

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-id 5549 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7430 df-oprab 7431 df-mpo 7432 df-1st 8008 df-2nd 8009 df-dir 18782 df-tail 18783 df-fbas 21687 df-fg 21688 df-fil 24177 df-fm 24269

This theorem is used by: (None)

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