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Theorem dprdval 16475
Description: The value of the internal direct product operation, which is a function mapping the (infinite, but finitely supported) cartesian product of subgroups (which mutually commute and have trivial intersections) to its (group) sum . (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
Hypotheses
Ref Expression
dprdval.0  |-  .0.  =  ( 0g `  G )
dprdval.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
Assertion
Ref Expression
dprdval  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) )
Distinct variable groups:    f, h, i, I    S, f, h, i    f, G, h, i
Allowed substitution hints:    W( f, h, i)    .0. ( f, h, i)

Proof of Theorem dprdval
Dummy variables  g 
s  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 454 . 2  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  G dom DProd  S )
2 reldmdprd 16469 . . . . . 6  |-  Rel  dom DProd
32brrelex2i 4876 . . . . 5  |-  ( G dom DProd  S  ->  S  e. 
_V )
43adantr 462 . . . 4  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  S  e.  _V )
52brrelexi 4875 . . . . . 6  |-  ( G dom DProd  s  ->  G  e.  _V )
6 breq1 4292 . . . . . . . 8  |-  ( g  =  G  ->  (
g dom DProd  s  <->  G dom DProd  s ) )
7 oveq1 6097 . . . . . . . . 9  |-  ( g  =  G  ->  (
g DProd  s )  =  ( G DProd  s ) )
8 fveq2 5688 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
9 dprdval.0 . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  G )
108, 9syl6eqr 2491 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
1110breq2d 4301 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
h finSupp  ( 0g `  g
)  <->  h finSupp  .0.  ) )
1211rabbidv 2962 . . . . . . . . . . 11  |-  ( g  =  G  ->  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  =  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  } )
13 oveq1 6097 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
g  gsumg  f )  =  ( G  gsumg  f ) )
1412, 13mpteq12dv 4367 . . . . . . . . . 10  |-  ( g  =  G  ->  (
f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  =  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  }  |->  ( G 
gsumg  f ) ) )
1514rneqd 5063 . . . . . . . . 9  |-  ( g  =  G  ->  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  |->  ( G  gsumg  f ) ) )
167, 15eqeq12d 2455 . . . . . . . 8  |-  ( g  =  G  ->  (
( g DProd  s )  =  ran  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  <->  ( G DProd  s )  =  ran  (
f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  |->  ( G  gsumg  f ) ) ) )
176, 16imbi12d 320 . . . . . . 7  |-  ( g  =  G  ->  (
( g dom DProd  s  -> 
( g DProd  s )  =  ran  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) ) )  <-> 
( G dom DProd  s  -> 
( G DProd  s )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  |->  ( G  gsumg  f ) ) ) ) )
18 df-br 4290 . . . . . . . . 9  |-  ( g dom DProd  s  <->  <. g ,  s >.  e.  dom DProd  )
19 fvex 5698 . . . . . . . . . . . . . . . . . 18  |-  ( s `
 i )  e. 
_V
2019rgenw 2781 . . . . . . . . . . . . . . . . 17  |-  A. i  e.  dom  s ( s `
 i )  e. 
_V
21 ixpexg 7283 . . . . . . . . . . . . . . . . 17  |-  ( A. i  e.  dom  s ( s `  i )  e.  _V  ->  X_ i  e.  dom  s ( s `
 i )  e. 
_V )
2220, 21ax-mp 5 . . . . . . . . . . . . . . . 16  |-  X_ i  e.  dom  s ( s `
 i )  e. 
_V
2322rabex 4440 . . . . . . . . . . . . . . 15  |-  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  e.  _V
2423mptex 5945 . . . . . . . . . . . . . 14  |-  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  e. 
_V
2524rnex 6511 . . . . . . . . . . . . 13  |-  ran  (
f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  e. 
_V
2625rgen2w 2782 . . . . . . . . . . . 12  |-  A. g  e.  Grp  A. s  e. 
{ h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) )  e. 
_V
27 df-dprd 16467 . . . . . . . . . . . . 13  |- DProd  =  ( g  e.  Grp , 
s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } 
|->  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )
2827fmpt2x 6639 . . . . . . . . . . . 12  |-  ( A. g  e.  Grp  A. s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) )  e. 
_V 
<-> DProd  : U_ g  e.  Grp  ( { g }  X.  { h  |  (
h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) --> _V )
2926, 28mpbi 208 . . . . . . . . . . 11  |- DProd  : U_ g  e.  Grp  ( { g }  X.  {
h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) --> _V
3029fdmi 5561 . . . . . . . . . 10  |-  dom DProd  =  U_ g  e.  Grp  ( { g }  X.  {
h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } )
3130eleq2i 2505 . . . . . . . . 9  |-  ( <.
g ,  s >.  e.  dom DProd 
<-> 
<. g ,  s >.  e.  U_ g  e.  Grp  ( { g }  X.  { h  |  (
h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) )
32 opeliunxp 4886 . . . . . . . . 9  |-  ( <.
g ,  s >.  e.  U_ g  e.  Grp  ( { g }  X.  { h  |  (
h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } )  <->  ( g  e. 
Grp  /\  s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) )
3318, 31, 323bitri 271 . . . . . . . 8  |-  ( g dom DProd  s  <->  ( g  e.  Grp  /\  s  e. 
{ h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) )
3427ovmpt4g 6212 . . . . . . . . 9  |-  ( ( g  e.  Grp  /\  s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) }  /\  ran  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  e. 
_V )  ->  (
g DProd  s )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )
3525, 34mp3an3 1298 . . . . . . . 8  |-  ( ( g  e.  Grp  /\  s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } )  ->  ( g DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )
3633, 35sylbi 195 . . . . . . 7  |-  ( g dom DProd  s  ->  (
g DProd  s )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )
3717, 36vtoclg 3027 . . . . . 6  |-  ( G  e.  _V  ->  ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  }  |->  ( G 
gsumg  f ) ) ) )
385, 37mpcom 36 . . . . 5  |-  ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  }  |->  ( G 
gsumg  f ) ) )
3938sbcth 3198 . . . 4  |-  ( S  e.  _V  ->  [. S  /  s ]. ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  }  |->  ( G 
gsumg  f ) ) ) )
404, 39syl 16 . . 3  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  [. S  /  s ]. ( G dom DProd  s  -> 
( G DProd  s )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  |->  ( G  gsumg  f ) ) ) )
41 simpr 458 . . . . . 6  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  s  =  S )
4241breq2d 4301 . . . . 5  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G dom DProd  s  <->  G dom DProd  S )
)
4341oveq2d 6106 . . . . . 6  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G DProd  s )  =  ( G DProd 
S ) )
4441dmeqd 5038 . . . . . . . . . . . . 13  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  s  =  dom  S )
45 simplr 749 . . . . . . . . . . . . 13  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  S  =  I )
4644, 45eqtrd 2473 . . . . . . . . . . . 12  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  s  =  I )
4746ixpeq1d 7271 . . . . . . . . . . 11  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e. 
dom  s ( s `
 i )  = 
X_ i  e.  I 
( s `  i
) )
4841fveq1d 5690 . . . . . . . . . . . 12  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( s `  i )  =  ( S `  i ) )
4948ixpeq2dv 7275 . . . . . . . . . . 11  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e.  I  ( s `  i )  =  X_ i  e.  I  ( S `  i )
)
5047, 49eqtrd 2473 . . . . . . . . . 10  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e. 
dom  s ( s `
 i )  = 
X_ i  e.  I 
( S `  i
) )
51 biidd 237 . . . . . . . . . 10  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( h finSupp  .0.  <->  h finSupp  .0.  ) )
5250, 51rabeqbidv 2965 . . . . . . . . 9  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  =  { h  e.  X_ i  e.  I 
( S `  i
)  |  h finSupp  .0.  } )
53 dprdval.w . . . . . . . . 9  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
5452, 53syl6eqr 2491 . . . . . . . 8  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  =  W )
55 eqidd 2442 . . . . . . . 8  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G  gsumg  f )  =  ( G 
gsumg  f ) )
5654, 55mpteq12dv 4367 . . . . . . 7  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  |->  ( G  gsumg  f ) )  =  ( f  e.  W  |->  ( G  gsumg  f ) ) )
5756rneqd 5063 . . . . . 6  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ran  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  |->  ( G  gsumg  f ) )  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) )
5843, 57eqeq12d 2455 . . . . 5  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  }  |->  ( G 
gsumg  f ) )  <->  ( G DProd  S )  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) ) )
5942, 58imbi12d 320 . . . 4  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  }  |->  ( G 
gsumg  f ) ) )  <-> 
( G dom DProd  S  -> 
( G DProd  S )  =  ran  ( f  e.  W  |->  ( G  gsumg  f ) ) ) ) )
604, 59sbcied 3220 . . 3  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( [. S  /  s ]. ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  }  |->  ( G 
gsumg  f ) ) )  <-> 
( G dom DProd  S  -> 
( G DProd  S )  =  ran  ( f  e.  W  |->  ( G  gsumg  f ) ) ) ) )
6140, 60mpbid 210 . 2  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G dom DProd  S  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) ) )
621, 61mpd 15 1  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   {cab 2427   A.wral 2713   {crab 2717   _Vcvv 2970   [.wsbc 3183    \ cdif 3322    i^i cin 3324    C_ wss 3325   {csn 3874   <.cop 3880   U.cuni 4088   U_ciun 4168   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   dom cdm 4836   ran crn 4837   "cima 4839   -->wf 5411   ` cfv 5415  (class class class)co 6090   X_cixp 7259   finSupp cfsupp 7616   0gc0g 14374    gsumg cgsu 14375  mrClscmrc 14517   Grpcgrp 15406  SubGrpcsubg 15668  Cntzccntz 15826   DProd cdprd 16465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-ixp 7260  df-dprd 16467
This theorem is referenced by:  eldprd  16476  dprdlub  16513
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