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Mirrors > Home > MPE Home > Th. List > fbsspw | Structured version Visualization version GIF version |
Description: A filter base on a set is a subset of the power set. (Contributed by Stefan O'Rear, 28-Jul-2015.) |
Ref | Expression |
---|---|
fbsspw | ⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐹 ⊆ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6933 | . . . 4 ⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐵 ∈ dom fBas) | |
2 | isfbas 23777 | . . . 4 ⊢ (𝐵 ∈ dom fBas → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))) |
4 | 3 | ibi 266 | . 2 ⊢ (𝐹 ∈ (fBas‘𝐵) → (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))) |
5 | 4 | simpld 493 | 1 ⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐹 ⊆ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2929 ∉ wnel 3035 ∀wral 3050 ∩ cin 3943 ⊆ wss 3944 ∅c0 4322 𝒫 cpw 4604 dom cdm 5678 ‘cfv 6549 fBascfbas 21284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fv 6557 df-fbas 21293 |
This theorem is referenced by: fbelss 23781 fbun 23788 filsspw 23799 fsubbas 23815 fgabs 23827 fmfnfm 23906 cfiluweak 24244 minveclem4a 25402 minveclem4 25404 |
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