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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 17534 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | |
2 | mresspw 17532 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋) | |
3 | elpwuni 5107 | . . 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 1541 ∈ wcel 2106 ⊆ wss 3947 𝒫 cpw 4601 ∪ cuni 4907 ‘cfv 6540 Moorecmre 17522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 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-iota 6492 df-fun 6542 df-fv 6548 df-mre 17526 |
This theorem is referenced by: mreunirn 17541 mrcfval 17548 mrcssv 17554 mrisval 17570 mrelatlub 18511 mreclatBAD 18512 mreuniss 47485 clduni 47486 mreclat 47575 |
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