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Mirrors > Home > MPE Home > Th. List > filunibas | Structured version Visualization version GIF version |
Description: Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.) |
Ref | Expression |
---|---|
filunibas | ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | filsspw 23284 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋) | |
2 | sspwuni 5096 | . . 3 ⊢ (𝐹 ⊆ 𝒫 𝑋 ↔ ∪ 𝐹 ⊆ 𝑋) | |
3 | 1, 2 | sylib 217 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 ⊆ 𝑋) |
4 | filtop 23288 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹) | |
5 | unissel 4935 | . 2 ⊢ ((∪ 𝐹 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐹) → ∪ 𝐹 = 𝑋) | |
6 | 3, 4, 5 | syl2anc 584 | 1 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⊆ wss 3944 𝒫 cpw 4596 ∪ cuni 4901 ‘cfv 6532 Filcfil 23278 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fv 6540 df-fbas 20875 df-fil 23279 |
This theorem is referenced by: filunirn 23315 filconn 23316 uffixfr 23356 uffix2 23357 uffixsn 23358 ufildr 23364 flimtopon 23403 flimss1 23406 flffval 23422 fclsval 23441 isfcls 23442 fclstopon 23445 fclsfnflim 23460 fcfval 23466 |
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