Theorem mrcf 14669

Description: The Moore closure is a function mapping arbitrary subsets to closed sets. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypothesis

Ref	Expression
mrcfval.f	|- F = (mrCls ` C )

Assertion

Ref	Expression
mrcf	|- ( C e. (Moore ` X ) -> F : ~P X --> C )

Proof of Theorem mrcf

Dummy variables x s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrcflem 14666	. 2 |- ( C e. (Moore ` X ) -> ( x e. ~P X |-> |^| { s e. C | x C_ s } ) : ~P X --> C )
2		mrcfval.f	. . . 4 |- F = (mrCls ` C )
3	2	mrcfval 14668	. . 3 |- ( C e. (Moore ` X ) -> F = ( x e. ~P X |-> |^| { s e. C | x C_ s } ) )
4	3	feq1d 5657	. 2 |- ( C e. (Moore ` X ) -> ( F : ~P X --> C <-> ( x e. ~P X |-> |^| { s e. C | x C_ s } ) : ~P X --> C ) )
5	1, 4	mpbird 232	1 |- ( C e. (Moore ` X ) -> F : ~P X --> C )

Syntax hints:   -> wi 4   = wceq 1370   e. wcel 1758   {crab 2803   C_ wss 3439   ~Pcpw 3971   |^|cint 4239   |-> cmpt 4461   -->wf 5525   ` cfv 5529  Moorecmre 14642  mrClscmrc 14643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-mre 14646  df-mrc 14647

This theorem is referenced by:  mrccl  14671  mrcssv  14674  mrcuni  14681  mrcun  14682  isacs2  14713  isacs4lem  15460  isacs5  15464  ismrcd2  29203  ismrc  29205  isnacs2  29210  isnacs3  29214