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Mirrors > Home > MPE Home > Th. List > filunirn | Structured version Visualization version GIF version |
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 6510 | . . . . . 6 ⊢ (fBas‘𝑦) ∈ V | |
2 | 1 | rabex 5088 | . . . . 5 ⊢ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧 ∈ 𝑤)} ∈ V |
3 | df-fil 22174 | . . . . 5 ⊢ Fil = (𝑦 ∈ V ↦ {𝑤 ∈ (fBas‘𝑦) ∣ ∀𝑧 ∈ 𝒫 𝑦((𝑤 ∩ 𝒫 𝑧) ≠ ∅ → 𝑧 ∈ 𝑤)}) | |
4 | 2, 3 | fnmpti 6319 | . . . 4 ⊢ Fil Fn V |
5 | fnunirn 6836 | . . . 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 22209 | . . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑥) → ∪ 𝐹 = 𝑥) | |
8 | 7 | fveq2d 6501 | . . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑥) → (Fil‘∪ 𝐹) = (Fil‘𝑥)) |
9 | 8 | eleq2d 2846 | . . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑥) → (𝐹 ∈ (Fil‘∪ 𝐹) ↔ 𝐹 ∈ (Fil‘𝑥))) |
10 | 9 | ibir 260 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘∪ 𝐹)) |
11 | 10 | rexlimivw 3222 | . . 3 ⊢ (∃𝑥 ∈ V 𝐹 ∈ (Fil‘𝑥) → 𝐹 ∈ (Fil‘∪ 𝐹)) |
12 | 6, 11 | sylbi 209 | . 2 ⊢ (𝐹 ∈ ∪ ran Fil → 𝐹 ∈ (Fil‘∪ 𝐹)) |
13 | fvssunirn 6526 | . . 3 ⊢ (Fil‘∪ 𝐹) ⊆ ∪ ran Fil | |
14 | 13 | sseli 3849 | . 2 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → 𝐹 ∈ ∪ ran Fil) |
15 | 12, 14 | impbii 201 | 1 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2051 ≠ wne 2962 ∀wral 3083 ∃wrex 3084 {crab 3087 Vcvv 3410 ∩ cin 3823 ∅c0 4173 𝒫 cpw 4417 ∪ cuni 4709 ran crn 5405 Fn wfn 6181 ‘cfv 6186 fBascfbas 20251 Filcfil 22173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-fv 6194 df-fbas 20260 df-fil 22174 |
This theorem is referenced by: flimfil 22297 isfcls 22337 |
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