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Mirrors > Home > MPE Home > Th. List > isfil | Structured version Visualization version GIF version |
Description: The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.) |
Ref | Expression |
---|---|
isfil | ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fil 22451 | . 2 ⊢ Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)}) | |
2 | pweq 4513 | . . . 4 ⊢ (𝑧 = 𝑋 → 𝒫 𝑧 = 𝒫 𝑋) | |
3 | 2 | adantr 484 | . . 3 ⊢ ((𝑧 = 𝑋 ∧ 𝑓 = 𝐹) → 𝒫 𝑧 = 𝒫 𝑋) |
4 | ineq1 4131 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝑥)) | |
5 | 4 | neeq1d 3046 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝑥) ≠ ∅)) |
6 | eleq2 2878 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ 𝑓 ↔ 𝑥 ∈ 𝐹)) | |
7 | 5, 6 | imbi12d 348 | . . . 4 ⊢ (𝑓 = 𝐹 → (((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓) ↔ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))) |
8 | 7 | adantl 485 | . . 3 ⊢ ((𝑧 = 𝑋 ∧ 𝑓 = 𝐹) → (((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓) ↔ ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))) |
9 | 3, 8 | raleqbidv 3354 | . 2 ⊢ ((𝑧 = 𝑋 ∧ 𝑓 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓) ↔ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))) |
10 | fveq2 6645 | . 2 ⊢ (𝑧 = 𝑋 → (fBas‘𝑧) = (fBas‘𝑋)) | |
11 | fvex 6658 | . 2 ⊢ (fBas‘𝑧) ∈ V | |
12 | elfvdm 6677 | . 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas) | |
13 | 1, 9, 10, 11, 12 | elmptrab2 22433 | 1 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∩ cin 3880 ∅c0 4243 𝒫 cpw 4497 dom cdm 5519 ‘cfv 6324 fBascfbas 20079 Filcfil 22450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 df-fil 22451 |
This theorem is referenced by: filfbas 22453 filss 22458 isfil2 22461 ustfilxp 22818 |
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