Theorem mrcflem 16266

Description: The domain and range of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Assertion

Ref	Expression
mrcflem	|- ( C e. (Moore ` X ) -> ( x e. ~P X |-> |^| { s e. C | x C_ s } ) : ~P X --> C )

Distinct variable groups:   x, s, C   x, X, s

Proof of Theorem mrcflem

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( C e. (Moore ` X ) /\ x e. ~P X ) -> C e. (Moore ` X ) )
2		ssrab2 3687	. . . 4 |- { s e. C | x C_ s } C_ C
3	2	a1i 11	. . 3 |- ( ( C e. (Moore ` X ) /\ x e. ~P X ) -> { s e. C | x C_ s } C_ C )
4		mre1cl 16254	. . . . . 6 |- ( C e. (Moore ` X ) -> X e. C )
5	4	adantr 481	. . . . 5 |- ( ( C e. (Moore ` X ) /\ x e. ~P X ) -> X e. C )
6		elpwi 4168	. . . . . 6 |- ( x e. ~P X -> x C_ X )
7	6	adantl 482	. . . . 5 |- ( ( C e. (Moore ` X ) /\ x e. ~P X ) -> x C_ X )
8		sseq2 3627	. . . . . 6 |- ( s = X -> ( x C_ s <-> x C_ X ) )
9	8	elrab 3363	. . . . 5 |- ( X e. { s e. C | x C_ s } <-> ( X e. C /\ x C_ X ) )
10	5, 7, 9	sylanbrc 698	. . . 4 |- ( ( C e. (Moore ` X ) /\ x e. ~P X ) -> X e. { s e. C | x C_ s } )
11		ne0i 3921	. . . 4 |- ( X e. { s e. C | x C_ s } -> { s e. C | x C_ s } =/= (/) )
12	10, 11	syl 17	. . 3 |- ( ( C e. (Moore ` X ) /\ x e. ~P X ) -> { s e. C | x C_ s } =/= (/) )
13		mreintcl 16255	. . 3 |- ( ( C e. (Moore ` X ) /\ { s e. C | x C_ s } C_ C /\ { s e. C | x C_ s } =/= (/) ) -> |^| { s e. C | x C_ s } e. C )
14	1, 3, 12, 13	syl3anc 1326	. 2 |- ( ( C e. (Moore ` X ) /\ x e. ~P X ) -> |^| { s e. C | x C_ s } e. C )
15		eqid 2622	. 2 |- ( x e. ~P X |-> |^| { s e. C | x C_ s } ) = ( x e. ~P X |-> |^| { s e. C | x C_ s } )
16	14, 15	fmptd 6385	1 |- ( C e. (Moore ` X ) -> ( x e. ~P X |-> |^| { s e. C | x C_ s } ) : ~P X --> C )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   =/= wne 2794   {crab 2916   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   |-> cmpt 4729   -->wf 5884   ` cfv 5888  Moorecmre 16242

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

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

This theorem is referenced by:  fnmrc  16267  mrcfval  16268  mrcf  16269