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Theorem reldmdprd 16897
Description: The domain of the internal direct product operation is a relation. (Contributed by Mario Carneiro, 25-Apr-2016.) (Proof shortened by AV, 11-Jul-2019.)
Assertion
Ref Expression
reldmdprd  |-  Rel  dom DProd

Proof of Theorem reldmdprd
Dummy variables  g  h  f  s  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dprd 16895 . 2  |- DProd  =  ( g  e.  Grp , 
s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  {
x } ) ( h `  x ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) } ) ) } 
|->  ran  ( f  e. 
{ h  e.  X_ x  e.  dom  s ( s `  x )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )
21reldmmpt2 6394 1  |-  Rel  dom DProd
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1381   {cab 2426   A.wral 2791   {crab 2795    \ cdif 3455    i^i cin 3457    C_ wss 3458   {csn 4010   U.cuni 4230   class class class wbr 4433    |-> cmpt 4491   dom cdm 4985   ran crn 4986   "cima 4988   Rel wrel 4990   -->wf 5570   ` cfv 5574  (class class class)co 6277   X_cixp 7467   finSupp cfsupp 7827   0gc0g 14709    gsumg cgsu 14710  mrClscmrc 14852   Grpcgrp 15922  SubGrpcsubg 16064  Cntzccntz 16222   DProd cdprd 16893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-br 4434  df-opab 4492  df-xp 4991  df-rel 4992  df-dm 4995  df-oprab 6281  df-mpt2 6282  df-dprd 16895
This theorem is referenced by:  dprddomprc  16900  dprdval0prc  16902  dprdval  16903  dprdvalOLD  16905  dprdgrp  16907  dprdf  16908  dprdwOLD  16918  dprdssv  16924  dprdfidOLD  16932  dprdfinvOLD  16934  dprdfaddOLD  16935  dprdfsubOLD  16936  dprdfeq0OLD  16937  dprdf11OLD  16938  subgdmdprd  16949  dmdprdsplitlemOLD  16953  dprddisj2OLD  16956  dprd2da  16959  dpjfval  16972  dpjidclOLD  16982
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