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Theorem mrcssv 16880
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 6678 . 2 (𝐹𝑈) ⊆ ran 𝐹
2 mrcfval.f . . . . 5 𝐹 = (mrCls‘𝐶)
32mrcf 16875 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
4 frn 6497 . . . 4 (𝐹:𝒫 𝑋𝐶 → ran 𝐹𝐶)
5 uniss 4811 . . . 4 (ran 𝐹𝐶 ran 𝐹 𝐶)
63, 4, 53syl 18 . . 3 (𝐶 ∈ (Moore‘𝑋) → ran 𝐹 𝐶)
7 mreuni 16866 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)
86, 7sseqtrd 3958 . 2 (𝐶 ∈ (Moore‘𝑋) → ran 𝐹𝑋)
91, 8sstrid 3929 1 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑈) ⊆ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  wss 3884  𝒫 cpw 4500   cuni 4803  ran crn 5524  wf 6324  cfv 6328  Moorecmre 16848  mrClscmrc 16849
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-mre 16852  df-mrc 16853
This theorem is referenced by:  mrcidb  16881  mrcuni  16887  mrcssvd  16889  mrefg2  39635  proot1hash  40131
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