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Theorem releqg 16864
Description: The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypothesis
Ref Expression
releqg.r  |-  R  =  ( G ~QG  S )
Assertion
Ref Expression
releqg  |-  Rel  R

Proof of Theorem releqg
Dummy variables  g 
s  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-eqg 16816 . . 3  |- ~QG  =  ( g  e.  _V ,  s  e. 
_V  |->  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  g )  /\  (
( ( invg `  g ) `  x
) ( +g  `  g
) y )  e.  s ) } )
21relmpt2opab 6878 . 2  |-  Rel  ( G ~QG  S )
3 releqg.r . . 3  |-  R  =  ( G ~QG  S )
43releqi 4918 . 2  |-  ( Rel 
R  <->  Rel  ( G ~QG  S ) )
52, 4mpbir 213 1  |-  Rel  R
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371    = wceq 1444    e. wcel 1887   _Vcvv 3045    C_ wss 3404   {cpr 3970   Rel wrel 4839   ` cfv 5582  (class class class)co 6290   Basecbs 15121   +g cplusg 15190   invgcminusg 16670   ~QG cqg 16813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-eqg 16816
This theorem is referenced by:  eqger  16867  eqgid  16869  tgptsmscls  21164
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