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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 23860 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋) | |
| 2 | sspwuni 5099 | . . 3 ⊢ (𝐹 ⊆ 𝒫 𝑋 ↔ ∪ 𝐹 ⊆ 𝑋) | |
| 3 | 1, 2 | sylib 218 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 ⊆ 𝑋) | 
| 4 | filtop 23864 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹) | |
| 5 | unissel 4937 | . 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 1539 ∈ wcel 2107 ⊆ wss 3950 𝒫 cpw 4599 ∪ cuni 4906 ‘cfv 6560 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-fv 6568 df-fbas 21362 df-fil 23855 | 
| This theorem is referenced by: filunirn 23891 filconn 23892 uffixfr 23932 uffix2 23933 uffixsn 23934 ufildr 23940 flimtopon 23979 flimss1 23982 flffval 23998 fclsval 24017 isfcls 24018 fclstopon 24021 fclsfnflim 24036 fcfval 24042 | 
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