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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 16462 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | |
2 | mresspw 16460 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋) | |
3 | elpwuni 4751 | . . 3 ⊢ (𝑋 ∈ 𝐶 → (𝐶 ⊆ 𝒫 𝑋 ↔ ∪ 𝐶 = 𝑋)) | |
4 | 3 | biimpa 462 | . 2 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐶 ⊆ 𝒫 𝑋) → ∪ 𝐶 = 𝑋) |
5 | 1, 2, 4 | syl2anc 573 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ⊆ wss 3723 𝒫 cpw 4298 ∪ cuni 4575 ‘cfv 6030 Moorecmre 16450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-iota 5993 df-fun 6032 df-fv 6038 df-mre 16454 |
This theorem is referenced by: mreunirn 16469 mrcfval 16476 mrcssv 16482 mrisval 16498 mrelatlub 17394 mreclatBAD 17395 |
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