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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 22459 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋) | |
2 | sspwuni 5022 | . . 3 ⊢ (𝐹 ⊆ 𝒫 𝑋 ↔ ∪ 𝐹 ⊆ 𝑋) | |
3 | 1, 2 | sylib 220 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 ⊆ 𝑋) |
4 | filtop 22463 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹) | |
5 | unissel 4869 | . 2 ⊢ ((∪ 𝐹 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐹) → ∪ 𝐹 = 𝑋) | |
6 | 3, 4, 5 | syl2anc 586 | 1 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 ‘cfv 6355 Filcfil 22453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fv 6363 df-fbas 20542 df-fil 22454 |
This theorem is referenced by: filunirn 22490 filconn 22491 uffixfr 22531 uffix2 22532 uffixsn 22533 ufildr 22539 flimtopon 22578 flimss1 22581 flffval 22597 fclsval 22616 isfcls 22617 fclstopon 22620 fclsfnflim 22635 fcfval 22641 |
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