Theorem reldmdprd 18396

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.

Step	Hyp	Ref	Expression
1		df-dprd 18394	. 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 ) ) )
2	1	reldmmpt2 6771	1 |- Rel dom DProd

Syntax hints:   /\ wa 384   = wceq 1483   {cab 2608   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   {csn 4177   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   "cima 5117   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   X_cixp 7908   finSupp cfsupp 8275   0gc0g 16100   gsum_g cgsu 16101  mrClscmrc 16243   Grpcgrp 17422  SubGrpcsubg 17588  Cntzccntz 17748   DProd cdprd 18392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124  df-oprab 6654  df-mpt2 6655  df-dprd 18394

This theorem is referenced by:  dprddomprc  18399  dprdval0prc  18401  dprdval  18402  dprdgrp  18404  dprdf  18405  dprdssv  18415  subgdmdprd  18433  dprd2da  18441  dpjfval  18454