Theorem mreuni 17510

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.)

Assertion

Ref	Expression
mreuni	⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)

Proof of Theorem mreuni

Step	Hyp	Ref	Expression
1		mre1cl 17504	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
2		mresspw 17502	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
3		elpwuni 5057	. . 3 ⊢ (𝑋 ∈ 𝐶 → (𝐶 ⊆ 𝒫 𝑋 ↔ ∪ 𝐶 = 𝑋))
4	3	biimpa 476	. 2 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐶 ⊆ 𝒫 𝑋) → ∪ 𝐶 = 𝑋)
5	1, 2, 4	syl2anc 584	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ⊆ wss 3898  𝒫 cpw 4551  ∪ cuni 4860  ‘cfv 6489  Moorecmre 17492

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-mre 17496

This theorem is referenced by:  mreunirn  17511  mrcfval  17522  mrcssv  17528  mrisval  17544  mrelatlub  18476  mreclatBAD  18477  mreuniss  49061  clduni  49062  mreclat  49158