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| Description: Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| filunirn | ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fvex 6918 | . . . . . 6 ⊢ (fBas‘𝑦) ∈ V | |
| 2 | 1 | rabex 5338 | . . . . 5 ⊢ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧 ∈ 𝑤)} ∈ V | 
| 3 | df-fil 23855 | . . . . 5 ⊢ Fil = (𝑦 ∈ V ↦ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧 ∈ 𝑤)}) | |
| 4 | 2, 3 | fnmpti 6710 | . . . 4 ⊢ Fil Fn V | 
| 5 | fnunirn 7275 | . . . 4 ⊢ (Fil Fn V → (𝐹 ∈ ∪ ran Fil ↔ ∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥))) | |
| 6 | 4, 5 | ax-mp 5 | . . 3 ⊢ (𝐹 ∈ ∪ ran Fil ↔ ∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥)) | 
| 7 | filunibas 23890 | . . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑥) → ∪ 𝐹 = 𝑥) | |
| 8 | 7 | fveq2d 6909 | . . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑥) → (Fil‘∪ 𝐹) = (Fil‘𝑥)) | 
| 9 | 8 | eleq2d 2826 | . . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑥) → (𝐹 ∈ (Fil‘∪ 𝐹) ↔ 𝐹 ∈ (Fil‘𝑥))) | 
| 10 | 9 | ibir 268 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘∪ 𝐹)) | 
| 11 | 10 | rexlimivw 3150 | . . 3 ⊢ (∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘∪ 𝐹)) | 
| 12 | 6, 11 | sylbi 217 | . 2 ⊢ (𝐹 ∈ ∪ ran Fil → 𝐹 ∈ (Fil‘∪ 𝐹)) | 
| 13 | fvssunirn 6938 | . . 3 ⊢ (Fil‘∪ 𝐹) ⊆ ∪ ran Fil | |
| 14 | 13 | sseli 3978 | . 2 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → 𝐹 ∈ ∪ ran Fil) | 
| 15 | 12, 14 | impbii 209 | 1 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3435 Vcvv 3479 ∩ cin 3949 ∅c0 4332 𝒫 cpw 4599 ∪ cuni 4906 ran crn 5685 Fn wfn 6555 ‘cfv 6560 fBascfbas 21353 Filcfil 23854 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-fv 6568 df-fbas 21362 df-fil 23855 | 
| This theorem is referenced by: flimfil 23978 isfcls 24018 | 
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