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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 21856 | . . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑡 ∈ 𝐹) → 𝑡 ⊆ 𝑋) | |
2 | 1 | ex 397 | . . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → 𝑡 ⊆ 𝑋)) |
3 | ssid 3773 | . . . . . 6 ⊢ 𝑡 ⊆ 𝑡 | |
4 | sseq1 3775 | . . . . . . 7 ⊢ (𝑥 = 𝑡 → (𝑥 ⊆ 𝑡 ↔ 𝑡 ⊆ 𝑡)) | |
5 | 4 | rspcev 3460 | . . . . . 6 ⊢ ((𝑡 ∈ 𝐹 ∧ 𝑡 ⊆ 𝑡) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡) |
6 | 3, 5 | mpan2 663 | . . . . 5 ⊢ (𝑡 ∈ 𝐹 → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡) |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)) |
8 | 2, 7 | jcad 496 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))) |
9 | elfg 21894 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))) | |
10 | 8, 9 | sylibrd 249 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → 𝑡 ∈ (𝑋filGen𝐹))) |
11 | 10 | ssrdv 3758 | 1 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∈ wcel 2145 ∃wrex 3062 ⊆ wss 3723 ‘cfv 6031 (class class class)co 6792 fBascfbas 19948 filGencfg 19949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fv 6039 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-fbas 19957 df-fg 19958 |
This theorem is referenced by: fgss2 21897 fgfil 21898 fgabs 21902 trfg 21914 isufil2 21931 ssufl 21941 ufileu 21942 filufint 21943 elfm2 21971 fmfnfmlem4 21980 fmfnfm 21981 fmco 21984 hausflim 22004 flimclslem 22007 flffbas 22018 fclsbas 22044 fclsfnflim 22050 flimfnfcls 22051 fclscmp 22053 isucn2 22302 cfilufg 22316 metust 22582 psmetutop 22591 fgcfil 23287 cmetss 23331 minveclem4a 23419 minveclem4 23421 fgmin 32699 |
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