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Theorem mrcssv 17491
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 6873 . 2 (𝐹𝑈) ⊆ ran 𝐹
2 mrcfval.f . . . . 5 𝐹 = (mrCls‘𝐶)
32mrcf 17486 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
4 frn 6673 . . . 4 (𝐹:𝒫 𝑋𝐶 → ran 𝐹𝐶)
5 uniss 4872 . . . 4 (ran 𝐹𝐶 ran 𝐹 𝐶)
63, 4, 53syl 18 . . 3 (𝐶 ∈ (Moore‘𝑋) → ran 𝐹 𝐶)
7 mreuni 17477 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)
86, 7sseqtrd 3983 . 2 (𝐶 ∈ (Moore‘𝑋) → ran 𝐹𝑋)
91, 8sstrid 3954 1 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑈) ⊆ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  wss 3909  𝒫 cpw 4559   cuni 4864  ran crn 5633  wf 6490  cfv 6494  Moorecmre 17459  mrClscmrc 17460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7669
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502  df-mre 17463  df-mrc 17464
This theorem is referenced by:  mrcidb  17492  mrcuni  17498  mrcssvd  17500  mrefg2  41006  proot1hash  41503
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