Theorem mrcf 15113

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 15110	. 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 15112	. . 3 |- ( C e. (Moore ` X ) -> F = ( x e. ~P X |-> |^| { s e. C | x C_ s } ) )
4	3	feq1d 5654	. 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 1403   e. wcel 1840   {crab 2755   C_ wss 3411   ~Pcpw 3952   |^|cint 4224   |-> cmpt 4450   -->wf 5519   ` cfv 5523  Moorecmre 15086  mrClscmrc 15087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-int 4225  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-fv 5531  df-mre 15090  df-mrc 15091

This theorem is referenced by:  mrccl  15115  mrcssv  15118  mrcuni  15125  mrcun  15126  isacs2  15157  isacs4lem  16012  isacs5  16016  ismrcd2  34957  ismrc  34959  isnacs2  34964  isnacs3  34968