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Theorem mrcssv 16097
 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 6127 . 2 (𝐹𝑈) ⊆ ran 𝐹
2 mrcfval.f . . . . 5 𝐹 = (mrCls‘𝐶)
32mrcf 16092 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
4 frn 5966 . . . 4 (𝐹:𝒫 𝑋𝐶 → ran 𝐹𝐶)
5 uniss 4394 . . . 4 (ran 𝐹𝐶 ran 𝐹 𝐶)
63, 4, 53syl 18 . . 3 (𝐶 ∈ (Moore‘𝑋) → ran 𝐹 𝐶)
7 mreuni 16083 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)
86, 7sseqtrd 3604 . 2 (𝐶 ∈ (Moore‘𝑋) → ran 𝐹𝑋)
91, 8syl5ss 3579 1 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑈) ⊆ 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ran crn 5039  ⟶wf 5800  ‘cfv 5804  Moorecmre 16065  mrClscmrc 16066 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-mre 16069  df-mrc 16070 This theorem is referenced by:  mrcidb  16098  mrcuni  16104  mrcssvd  16106  mrefg2  36288  proot1hash  36797
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