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| 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 23842 | . . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑡 ∈ 𝐹) → 𝑡 ⊆ 𝑋) | |
| 2 | 1 | ex 412 | . . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → 𝑡 ⊆ 𝑋)) | 
| 3 | ssid 4005 | . . . . 5 ⊢ 𝑡 ⊆ 𝑡 | |
| 4 | sseq1 4008 | . . . . . 6 ⊢ (𝑥 = 𝑡 → (𝑥 ⊆ 𝑡 ↔ 𝑡 ⊆ 𝑡)) | |
| 5 | 4 | rspcev 3621 | . . . . 5 ⊢ ((𝑡 ∈ 𝐹 ∧ 𝑡 ⊆ 𝑡) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡) | 
| 6 | 3, 5 | mpan2 691 | . . . 4 ⊢ (𝑡 ∈ 𝐹 → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡) | 
| 7 | 2, 6 | jca2 513 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))) | 
| 8 | elfg 23880 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))) | |
| 9 | 7, 8 | sylibrd 259 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → 𝑡 ∈ (𝑋filGen𝐹))) | 
| 10 | 9 | ssrdv 3988 | 1 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 ∃wrex 3069 ⊆ wss 3950 ‘cfv 6560 (class class class)co 7432 fBascfbas 21353 filGencfg 21354 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-fbas 21362 df-fg 21363 | 
| This theorem is referenced by: fgss2 23883 fgfil 23884 fgabs 23888 trfg 23900 isufil2 23917 ssufl 23927 ufileu 23928 filufint 23929 elfm2 23957 fmfnfmlem4 23966 fmfnfm 23967 fmco 23970 hausflim 23990 flimclslem 23993 flffbas 24004 fclsbas 24030 fclsfnflim 24036 flimfnfcls 24037 fclscmp 24039 isucn2 24289 cfilufg 24303 metust 24572 psmetutop 24581 fgcfil 25306 cmetss 25351 minveclem4a 25465 minveclem4 25467 fgmin 36372 | 
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