Theorem mrcssv 17582

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 6894	. 2 ⊢ (𝐹‘𝑈) ⊆ ∪ ran 𝐹
2		mrcfval.f	. . . . 5 ⊢ 𝐹 = (mrCls‘𝐶)
3	2	mrcf 17577	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
4		frn 6698	. . . 4 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → ran 𝐹 ⊆ 𝐶)
5		uniss 4882	. . . 4 ⊢ (ran 𝐹 ⊆ 𝐶 → ∪ ran 𝐹 ⊆ ∪ 𝐶)
6	3, 4, 5	3syl 18	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ ran 𝐹 ⊆ ∪ 𝐶)
7		mreuni 17568	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
8	6, 7	sseqtrd 3986	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ ran 𝐹 ⊆ 𝑋)
9	1, 8	sstrid 3961	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑈) ⊆ 𝑋)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ⊆ wss 3917  𝒫 cpw 4566  ∪ cuni 4874  ran crn 5642  ⟶wf 6510  ‘cfv 6514  Moorecmre 17550  mrClscmrc 17551

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-mre 17554  df-mrc 17555

This theorem is referenced by:  mrcidb  17583  mrcuni  17589  mrcssvd  17591  mrefg2  42702  proot1hash  43191