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Mirrors > Home > MPE Home > Th. List > ssfg | Structured version Visualization version GIF version |
Description: A filter base is a subset of its generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
Ref | Expression |
---|---|
ssfg | ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fbelss 23266 | . . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑡 ∈ 𝐹) → 𝑡 ⊆ 𝑋) | |
2 | 1 | ex 413 | . . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → 𝑡 ⊆ 𝑋)) |
3 | ssid 4000 | . . . . 5 ⊢ 𝑡 ⊆ 𝑡 | |
4 | sseq1 4003 | . . . . . 6 ⊢ (𝑥 = 𝑡 → (𝑥 ⊆ 𝑡 ↔ 𝑡 ⊆ 𝑡)) | |
5 | 4 | rspcev 3609 | . . . . 5 ⊢ ((𝑡 ∈ 𝐹 ∧ 𝑡 ⊆ 𝑡) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡) |
6 | 3, 5 | mpan2 689 | . . . 4 ⊢ (𝑡 ∈ 𝐹 → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡) |
7 | 2, 6 | jca2 514 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))) |
8 | elfg 23304 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))) | |
9 | 7, 8 | sylibrd 258 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → 𝑡 ∈ (𝑋filGen𝐹))) |
10 | 9 | ssrdv 3984 | 1 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∃wrex 3069 ⊆ wss 3944 ‘cfv 6532 (class class class)co 7393 fBascfbas 20866 filGencfg 20867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-fbas 20875 df-fg 20876 |
This theorem is referenced by: fgss2 23307 fgfil 23308 fgabs 23312 trfg 23324 isufil2 23341 ssufl 23351 ufileu 23352 filufint 23353 elfm2 23381 fmfnfmlem4 23390 fmfnfm 23391 fmco 23394 hausflim 23414 flimclslem 23417 flffbas 23428 fclsbas 23454 fclsfnflim 23460 flimfnfcls 23461 fclscmp 23463 isucn2 23713 cfilufg 23727 metust 23996 psmetutop 24005 fgcfil 24717 cmetss 24762 minveclem4a 24876 minveclem4 24878 fgmin 35057 |
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