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Mirrors > Home > MPE Home > Th. List > mreuni | Structured version Visualization version GIF version |
Description: Since the entire base set of a Moore collection is the greatest element of it, the base set can be recovered from a Moore collection by set union. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
mreuni | ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mre1cl 16853 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | |
2 | mresspw 16851 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋) | |
3 | elpwuni 5018 | . . 3 ⊢ (𝑋 ∈ 𝐶 → (𝐶 ⊆ 𝒫 𝑋 ↔ ∪ 𝐶 = 𝑋)) | |
4 | 3 | biimpa 477 | . 2 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐶 ⊆ 𝒫 𝑋) → ∪ 𝐶 = 𝑋) |
5 | 1, 2, 4 | syl2anc 584 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 ‘cfv 6348 Moorecmre 16841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-mre 16845 |
This theorem is referenced by: mreunirn 16860 mrcfval 16867 mrcssv 16873 mrisval 16889 mrelatlub 17784 mreclatBAD 17785 |
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