Theorem mrcssv 17727

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 6912	. 2 ⊢ (𝐹‘𝑈) ⊆ ∪ ran 𝐹
2		mrcfval.f	. . . . 5 ⊢ 𝐹 = (mrCls‘𝐶)
3	2	mrcf 17722	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
4		frn 6713	. . . 4 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → ran 𝐹 ⊆ 𝐶)
5		uniss 4875	. . . 4 ⊢ (ran 𝐹 ⊆ 𝐶 → ∪ ran 𝐹 ⊆ ∪ 𝐶)
6	3, 4, 5	3syl 19	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ ran 𝐹 ⊆ ∪ 𝐶)
7		mreuni 17709	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋)
8	6, 7	sseqtrd 3967	. 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ ran 𝐹 ⊆ 𝑋)
9	1, 8	sstrid 3942	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑈) ⊆ 𝑋)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  ran crn 5656  ⟶wf 6531  ‘cfv 6535  Moorecmre 17691  mrClscmrc 17692

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-fv 6543  df-mre 17695  df-mrc 17696

This theorem is used by:  mrcidb  17728  mrcuni  17734  mrcssvd  17736  mrefg2  43617  proot1hash  44101