Theorem fnmrc 16267

Description: Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
fnmrc	|- mrCls Fn U. ran Moore

Proof of Theorem fnmrc

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

Step	Hyp	Ref	Expression
1		df-mrc 16247	. . 3 |- mrCls = ( c e. U. ran Moore |-> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) )
2	1	fnmpt 6020	. 2 |- ( A. c e. U. ran Moore ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) e. _V -> mrCls Fn U. ran Moore )
3		mreunirn 16261	. . 3 |- ( c e. U. ran Moore <-> c e. (Moore ` U. c ) )
4		mrcflem 16266	. . . . 5 |- ( c e. (Moore ` U. c ) -> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) : ~P U. c --> c )
5		fssxp 6060	. . . . 5 |- ( ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) : ~P U. c --> c -> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) C_ ( ~P U. c X. c ) )
6	4, 5	syl 17	. . . 4 |- ( c e. (Moore ` U. c ) -> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) C_ ( ~P U. c X. c ) )
7		vuniex 6954	. . . . . 6 |- U. c e. _V
8	7	pwex 4848	. . . . 5 |- ~P U. c e. _V
9		vex 3203	. . . . 5 |- c e. _V
10	8, 9	xpex 6962	. . . 4 |- ( ~P U. c X. c ) e. _V
11		ssexg 4804	. . . 4 |- ( ( ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) C_ ( ~P U. c X. c ) /\ ( ~P U. c X. c ) e. _V ) -> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) e. _V )
12	6, 10, 11	sylancl 694	. . 3 |- ( c e. (Moore ` U. c ) -> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) e. _V )
13	3, 12	sylbi 207	. 2 |- ( c e. U. ran Moore -> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) e. _V )
14	2, 13	mprg 2926	1 |- mrCls Fn U. ran Moore

Syntax hints:   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   |-> cmpt 4729   X. cxp 5112   ran crn 5115   Fn wfn 5883   -->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:  ismrc  37264