MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mrcssv Structured version   Visualization version   GIF version

Theorem mrcssv 17657
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) → (𝐹𝑈) ⊆ 𝑋)

Proof of Theorem mrcssv
StepHypRef Expression
1 fvssunirn 6939 . 2 (𝐹𝑈) ⊆ ran 𝐹
2 mrcfval.f . . . . 5 𝐹 = (mrCls‘𝐶)
32mrcf 17652 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
4 frn 6743 . . . 4 (𝐹:𝒫 𝑋𝐶 → ran 𝐹𝐶)
5 uniss 4915 . . . 4 (ran 𝐹𝐶 ran 𝐹 𝐶)
63, 4, 53syl 18 . . 3 (𝐶 ∈ (Moore‘𝑋) → ran 𝐹 𝐶)
7 mreuni 17643 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)
86, 7sseqtrd 4020 . 2 (𝐶 ∈ (Moore‘𝑋) → ran 𝐹𝑋)
91, 8sstrid 3995 1 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑈) ⊆ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wss 3951  𝒫 cpw 4600   cuni 4907  ran crn 5686  wf 6557  cfv 6561  Moorecmre 17625  mrClscmrc 17626
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  ax-un 7755
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-ne 2941  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-int 4947  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-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mre 17629  df-mrc 17630
This theorem is referenced by:  mrcidb  17658  mrcuni  17664  mrcssvd  17666  mrefg2  42718  proot1hash  43207
  Copyright terms: Public domain W3C validator