Theorem cntzex 14144

Description: Set existence of the centralizer. (Contributed by Jim Kingdon, 15-Sep-2026.)

Hypothesis

Ref	Expression
cntzex.z	|- Z = (Cntz ` M )

Assertion

Ref	Expression
cntzex	|- ( M e. V -> Z e. _V )

Proof of Theorem cntzex

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

Step	Hyp	Ref	Expression
1		cntzex.z	. 2 |- Z = (Cntz ` M )
2		df-cntz 14142	. . . 4 |- Cntz = ( m e. _V |-> ( s e. ~P ( Base ` m ) |-> { x e. ( Base ` m ) | A. y e. s ( x ( +g ` m ) y ) = ( y ( +g ` m ) x ) } ) )
3		fveq2 5695	. . . . . 6 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
4	3	pweqd 3693	. . . . 5 |- ( m = M -> ~P ( Base ` m ) = ~P ( Base ` M ) )
5		fveq2 5695	. . . . . . . . 9 |- ( m = M -> ( +g ` m ) = ( +g ` M ) )
6	5	oveqd 6102	. . . . . . . 8 |- ( m = M -> ( x ( +g ` m ) y ) = ( x ( +g ` M ) y ) )
7	5	oveqd 6102	. . . . . . . 8 |- ( m = M -> ( y ( +g ` m ) x ) = ( y ( +g ` M ) x ) )
8	6, 7	eqeq12d 2253	. . . . . . 7 |- ( m = M -> ( ( x ( +g ` m ) y ) = ( y ( +g ` m ) x ) <-> ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) ) )
9	8	ralbidv 2550	. . . . . 6 |- ( m = M -> ( A. y e. s ( x ( +g ` m ) y ) = ( y ( +g ` m ) x ) <-> A. y e. s ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) ) )
10	3, 9	rabeqbidv 2816	. . . . 5 |- ( m = M -> { x e. ( Base ` m ) | A. y e. s ( x ( +g ` m ) y ) = ( y ( +g ` m ) x ) } = { x e. ( Base ` M ) | A. y e. s ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) } )
11	4, 10	mpteq12dv 4213	. . . 4 |- ( m = M -> ( s e. ~P ( Base ` m ) |-> { x e. ( Base ` m ) | A. y e. s ( x ( +g ` m ) y ) = ( y ( +g ` m ) x ) } ) = ( s e. ~P ( Base ` M ) |-> { x e. ( Base ` M ) | A. y e. s ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) } ) )
12		elex 2833	. . . 4 |- ( M e. V -> M e. _V )
13		basfn 13463	. . . . . . 7 |- Base Fn _V
14		funfvex 5712	. . . . . . . 8 |- ( ( Fun Base /\ M e. dom Base ) -> ( Base ` M ) e. _V )
15	14	funfni 5483	. . . . . . 7 |- ( ( Base Fn _V /\ M e. _V ) -> ( Base ` M ) e. _V )
16	13, 12, 15	sylancr 418	. . . . . 6 |- ( M e. V -> ( Base ` M ) e. _V )
17	16	pwexd 4318	. . . . 5 |- ( M e. V -> ~P ( Base ` M ) e. _V )
18	17	mptexd 5944	. . . 4 |- ( M e. V -> ( s e. ~P ( Base ` M ) |-> { x e. ( Base ` M ) | A. y e. s ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) } ) e. _V )
19	2, 11, 12, 18	fvmptd3 5799	. . 3 |- ( M e. V -> (Cntz ` M ) = ( s e. ~P ( Base ` M ) |-> { x e. ( Base ` M ) | A. y e. s ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) } ) )
20	19, 18	eqeltrd 2315	. 2 |- ( M e. V -> (Cntz ` M ) e. _V )
21	1, 20	eqeltrid 2325	1 |- ( M e. V -> Z e. _V )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   ~Pcpw 3688   |-> cmpt 4192   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   Basecbs 13404   +g cplusg 13484  Cntzccntz 14140

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8271  ax-resscn 8272  ax-1re 8274  ax-addrcl 8277

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-inn 9308  df-ndx 13407  df-slot 13408  df-base 13410  df-cntz 14142

This theorem is used by:  cntrval  14145