MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reldmdprd Structured version   Unicode version

Theorem reldmdprd 17223
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 17221 . 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 6386 1  |-  Rel  dom DProd
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398   {cab 2439   A.wral 2804   {crab 2808    \ cdif 3458    i^i cin 3460    C_ wss 3461   {csn 4016   U.cuni 4235   class class class wbr 4439    |-> cmpt 4497   dom cdm 4988   ran crn 4989   "cima 4991   Rel wrel 4993   -->wf 5566   ` cfv 5570  (class class class)co 6270   X_cixp 7462   finSupp cfsupp 7821   0gc0g 14929    gsumg cgsu 14930  mrClscmrc 15072   Grpcgrp 16252  SubGrpcsubg 16394  Cntzccntz 16552   DProd cdprd 17219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-dm 4998  df-oprab 6274  df-mpt2 6275  df-dprd 17221
This theorem is referenced by:  dprddomprc  17226  dprdval0prc  17228  dprdval  17229  dprdvalOLD  17231  dprdgrp  17233  dprdf  17234  dprdwOLD  17245  dprdssv  17251  dprdfidOLD  17259  dprdfinvOLD  17261  dprdfaddOLD  17262  dprdfsubOLD  17263  dprdfeq0OLD  17264  dprdf11OLD  17265  subgdmdprd  17276  dmdprdsplitlemOLD  17280  dprddisj2OLD  17283  dprd2da  17286  dpjfval  17299  dpjidclOLD  17309
  Copyright terms: Public domain W3C validator