Theorem mrcssv 17659

Description: The closure of a set is a subset of the base. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypothesis

Ref	Expression
mrcfval.f	⊢ 𝐹 = (mrCls‘𝐶)

Assertion

Ref	Expression
mrcssv	⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑈) ⊆ 𝑋)

Proof of Theorem mrcssv

Step	Hyp	Ref	Expression
1		fvssunirn 6940	. 2 ⊢ (𝐹‘𝑈) ⊆ ∪ ran 𝐹
2		mrcfval.f	. . . . 5 ⊢ 𝐹 = (mrCls‘𝐶)
3	2	mrcf 17654	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
4		frn 6744	. . . 4 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → ran 𝐹 ⊆ 𝐶)
5		uniss 4920	. . . 4 ⊢ (ran 𝐹 ⊆ 𝐶 → ∪ ran 𝐹 ⊆ ∪ 𝐶)
6	3, 4, 5	3syl 18	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ ran 𝐹 ⊆ ∪ 𝐶)
7		mreuni 17645	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
8	6, 7	sseqtrd 4036	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ ran 𝐹 ⊆ 𝑋)
9	1, 8	sstrid 4007	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑈) ⊆ 𝑋)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106   ⊆ wss 3963  𝒫 cpw 4605  ∪ cuni 4912  ran crn 5690  ⟶wf 6559  ‘cfv 6563  Moorecmre 17627  mrClscmrc 17628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-mre 17631  df-mrc 17632

This theorem is referenced by:  mrcidb  17660  mrcuni  17666  mrcssvd  17668  mrefg2  42695  proot1hash  43184