Theorem mreuni 17643

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 17637	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
2		mresspw 17635	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
3		elpwuni 5105	. . 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 1540   ∈ wcel 2108   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907  ‘cfv 6561  Moorecmre 17625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-mre 17629

This theorem is referenced by:  mreunirn  17644  mrcfval  17651  mrcssv  17657  mrisval  17673  mrelatlub  18607  mreclatBAD  18608  mreuniss  48797  clduni  48798  mreclat  48886