Theorem cntzrcl 17760

Description: Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)

Hypotheses

Ref	Expression
cntzrcl.b	|- B = ( Base ` M )
cntzrcl.z	|- Z = (Cntz ` M )

Assertion

Ref	Expression
cntzrcl	|- ( X e. ( Z ` S ) -> ( M e. _V /\ S C_ B ) )

Proof of Theorem cntzrcl

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3919	. . . 4 |- -. X e. (/)
2		cntzrcl.z	. . . . . . . 8 |- Z = (Cntz ` M )
3		fvprc 6185	. . . . . . . 8 |- ( -. M e. _V -> (Cntz ` M ) = (/) )
4	2, 3	syl5eq 2668	. . . . . . 7 |- ( -. M e. _V -> Z = (/) )
5	4	fveq1d 6193	. . . . . 6 |- ( -. M e. _V -> ( Z ` S ) = ( (/) ` S ) )
6		0fv 6227	. . . . . 6 |- ( (/) ` S ) = (/)
7	5, 6	syl6eq 2672	. . . . 5 |- ( -. M e. _V -> ( Z ` S ) = (/) )
8	7	eleq2d 2687	. . . 4 |- ( -. M e. _V -> ( X e. ( Z ` S ) <-> X e. (/) ) )
9	1, 8	mtbiri 317	. . 3 |- ( -. M e. _V -> -. X e. ( Z ` S ) )
10	9	con4i 113	. 2 |- ( X e. ( Z ` S ) -> M e. _V )
11		cntzrcl.b	. . . . . . . 8 |- B = ( Base ` M )
12		eqid 2622	. . . . . . . 8 |- ( +g ` M ) = ( +g ` M )
13	11, 12, 2	cntzfval 17753	. . . . . . 7 |- ( M e. _V -> Z = ( x e. ~P B |-> { y e. B | A. z e. x ( y ( +g ` M ) z ) = ( z ( +g ` M ) y ) } ) )
14	10, 13	syl 17	. . . . . 6 |- ( X e. ( Z ` S ) -> Z = ( x e. ~P B |-> { y e. B | A. z e. x ( y ( +g ` M ) z ) = ( z ( +g ` M ) y ) } ) )
15	14	dmeqd 5326	. . . . 5 |- ( X e. ( Z ` S ) -> dom Z = dom ( x e. ~P B |-> { y e. B | A. z e. x ( y ( +g ` M ) z ) = ( z ( +g ` M ) y ) } ) )
16		eqid 2622	. . . . . 6 |- ( x e. ~P B |-> { y e. B | A. z e. x ( y ( +g ` M ) z ) = ( z ( +g ` M ) y ) } ) = ( x e. ~P B |-> { y e. B | A. z e. x ( y ( +g ` M ) z ) = ( z ( +g ` M ) y ) } )
17	16	dmmptss 5631	. . . . 5 |- dom ( x e. ~P B |-> { y e. B | A. z e. x ( y ( +g ` M ) z ) = ( z ( +g ` M ) y ) } ) C_ ~P B
18	15, 17	syl6eqss 3655	. . . 4 |- ( X e. ( Z ` S ) -> dom Z C_ ~P B )
19		elfvdm 6220	. . . 4 |- ( X e. ( Z ` S ) -> S e. dom Z )
20	18, 19	sseldd 3604	. . 3 |- ( X e. ( Z ` S ) -> S e. ~P B )
21	20	elpwid 4170	. 2 |- ( X e. ( Z ` S ) -> S C_ B )
22	10, 21	jca 554	1 |- ( X e. ( Z ` S ) -> ( M e. _V /\ S C_ B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Cntzccntz 17748

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-cntz 17750

This theorem is referenced by:  cntzssv  17761  cntzi  17762  resscntz  17764  cntzmhm  17771  oppgcntz  17794