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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 23575 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋) | |
2 | sspwuni 5102 | . . 3 ⊢ (𝐹 ⊆ 𝒫 𝑋 ↔ ∪ 𝐹 ⊆ 𝑋) | |
3 | 1, 2 | sylib 217 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 ⊆ 𝑋) |
4 | filtop 23579 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹) | |
5 | unissel 4941 | . 2 ⊢ ((∪ 𝐹 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐹) → ∪ 𝐹 = 𝑋) | |
6 | 3, 4, 5 | syl2anc 582 | 1 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 ⊆ wss 3947 𝒫 cpw 4601 ∪ cuni 4907 ‘cfv 6542 Filcfil 23569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fv 6550 df-fbas 21141 df-fil 23570 |
This theorem is referenced by: filunirn 23606 filconn 23607 uffixfr 23647 uffix2 23648 uffixsn 23649 ufildr 23655 flimtopon 23694 flimss1 23697 flffval 23713 fclsval 23732 isfcls 23733 fclstopon 23736 fclsfnflim 23751 fcfval 23757 |
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