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Theorem cntzsnval 14849
Description: Special substitution for the centralizer of a singleton. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
cntzfval.b  |-  B  =  ( Base `  M
)
cntzfval.p  |-  .+  =  ( +g  `  M )
cntzfval.z  |-  Z  =  (Cntz `  M )
Assertion
Ref Expression
cntzsnval  |-  ( Y  e.  B  ->  ( Z `  { Y } )  =  {
x  e.  B  | 
( x  .+  Y
)  =  ( Y 
.+  x ) } )
Distinct variable groups:    x,  .+    x, B    x, M    x, Y
Allowed substitution hint:    Z( x)

Proof of Theorem cntzsnval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 snssi 3796 . . 3  |-  ( Y  e.  B  ->  { Y }  C_  B )
2 cntzfval.b . . . 4  |-  B  =  ( Base `  M
)
3 cntzfval.p . . . 4  |-  .+  =  ( +g  `  M )
4 cntzfval.z . . . 4  |-  Z  =  (Cntz `  M )
52, 3, 4cntzval 14846 . . 3  |-  ( { Y }  C_  B  ->  ( Z `  { Y } )  =  {
x  e.  B  |  A. y  e.  { Y }  ( x  .+  y )  =  ( y  .+  x ) } )
61, 5syl 15 . 2  |-  ( Y  e.  B  ->  ( Z `  { Y } )  =  {
x  e.  B  |  A. y  e.  { Y }  ( x  .+  y )  =  ( y  .+  x ) } )
7 oveq2 5908 . . . . 5  |-  ( y  =  Y  ->  (
x  .+  y )  =  ( x  .+  Y ) )
8 oveq1 5907 . . . . 5  |-  ( y  =  Y  ->  (
y  .+  x )  =  ( Y  .+  x ) )
97, 8eqeq12d 2330 . . . 4  |-  ( y  =  Y  ->  (
( x  .+  y
)  =  ( y 
.+  x )  <->  ( x  .+  Y )  =  ( Y  .+  x ) ) )
109ralsng 3706 . . 3  |-  ( Y  e.  B  ->  ( A. y  e.  { Y }  ( x  .+  y )  =  ( y  .+  x )  <-> 
( x  .+  Y
)  =  ( Y 
.+  x ) ) )
1110rabbidv 2814 . 2  |-  ( Y  e.  B  ->  { x  e.  B  |  A. y  e.  { Y }  ( x  .+  y )  =  ( y  .+  x ) }  =  { x  e.  B  |  (
x  .+  Y )  =  ( Y  .+  x ) } )
126, 11eqtrd 2348 1  |-  ( Y  e.  B  ->  ( Z `  { Y } )  =  {
x  e.  B  | 
( x  .+  Y
)  =  ( Y 
.+  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701   A.wral 2577   {crab 2581    C_ wss 3186   {csn 3674   ` cfv 5292  (class class class)co 5900   Basecbs 13195   +g cplusg 13255  Cntzccntz 14840
This theorem is referenced by:  elcntzsn  14850  cntziinsn  14859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-cntz 14842
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