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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 22441 | . . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑡 ∈ 𝐹) → 𝑡 ⊆ 𝑋) | |
2 | 1 | ex 415 | . . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → 𝑡 ⊆ 𝑋)) |
3 | ssid 3989 | . . . . 5 ⊢ 𝑡 ⊆ 𝑡 | |
4 | sseq1 3992 | . . . . . 6 ⊢ (𝑥 = 𝑡 → (𝑥 ⊆ 𝑡 ↔ 𝑡 ⊆ 𝑡)) | |
5 | 4 | rspcev 3623 | . . . . 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 516 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))) |
8 | elfg 22479 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))) | |
9 | 7, 8 | sylibrd 261 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → 𝑡 ∈ (𝑋filGen𝐹))) |
10 | 9 | ssrdv 3973 | 1 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ∃wrex 3139 ⊆ wss 3936 ‘cfv 6355 (class class class)co 7156 fBascfbas 20533 filGencfg 20534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-fbas 20542 df-fg 20543 |
This theorem is referenced by: fgss2 22482 fgfil 22483 fgabs 22487 trfg 22499 isufil2 22516 ssufl 22526 ufileu 22527 filufint 22528 elfm2 22556 fmfnfmlem4 22565 fmfnfm 22566 fmco 22569 hausflim 22589 flimclslem 22592 flffbas 22603 fclsbas 22629 fclsfnflim 22635 flimfnfcls 22636 fclscmp 22638 isucn2 22888 cfilufg 22902 metust 23168 psmetutop 23177 fgcfil 23874 cmetss 23919 minveclem4a 24033 minveclem4 24035 fgmin 33718 |
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