Theorem mrcssv 16274

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 e. (Moore ` X ) -> ( F ` U ) C_ X )

Proof of Theorem mrcssv

Step	Hyp	Ref	Expression
1		fvssunirn 6217	. 2 |- ( F ` U ) C_ U. ran F
2		mrcfval.f	. . . . 5 |- F = (mrCls ` C )
3	2	mrcf 16269	. . . 4 |- ( C e. (Moore ` X ) -> F : ~P X --> C )
4		frn 6053	. . . 4 |- ( F : ~P X --> C -> ran F C_ C )
5		uniss 4458	. . . 4 |- ( ran F C_ C -> U. ran F C_ U. C )
6	3, 4, 5	3syl 18	. . 3 |- ( C e. (Moore ` X ) -> U. ran F C_ U. C )
7		mreuni 16260	. . 3 |- ( C e. (Moore ` X ) -> U. C = X )
8	6, 7	sseqtrd 3641	. 2 |- ( C e. (Moore ` X ) -> U. ran F C_ X )
9	1, 8	syl5ss 3614	1 |- ( C e. (Moore ` X ) -> ( F ` U ) C_ X )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   ran crn 5115   -->wf 5884   ` cfv 5888  Moorecmre 16242  mrClscmrc 16243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-mre 16246  df-mrc 16247

This theorem is referenced by:  mrcidb  16275  mrcuni  16281  mrcssvd  16283  mrefg2  37270  proot1hash  37778