![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 23857 | . . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑡 ∈ 𝐹) → 𝑡 ⊆ 𝑋) | |
2 | 1 | ex 412 | . . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ 𝐹 → 𝑡 ⊆ 𝑋)) |
3 | ssid 4018 | . . . . 5 ⊢ 𝑡 ⊆ 𝑡 | |
4 | sseq1 4021 | . . . . . 6 ⊢ (𝑥 = 𝑡 → (𝑥 ⊆ 𝑡 ↔ 𝑡 ⊆ 𝑡)) | |
5 | 4 | rspcev 3622 | . . . . 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 23895 | . . 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 4001 | 1 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 fBascfbas 21370 filGencfg 21371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-fbas 21379 df-fg 21380 |
This theorem is referenced by: fgss2 23898 fgfil 23899 fgabs 23903 trfg 23915 isufil2 23932 ssufl 23942 ufileu 23943 filufint 23944 elfm2 23972 fmfnfmlem4 23981 fmfnfm 23982 fmco 23985 hausflim 24005 flimclslem 24008 flffbas 24019 fclsbas 24045 fclsfnflim 24051 flimfnfcls 24052 fclscmp 24054 isucn2 24304 cfilufg 24318 metust 24587 psmetutop 24596 fgcfil 25319 cmetss 25364 minveclem4a 25478 minveclem4 25480 fgmin 36353 |
Copyright terms: Public domain | W3C validator |