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Theorem releqg 14713
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 14669 . . 3  |- ~QG  =  ( g  e.  _V ,  s  e. 
_V  |->  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  g )  /\  (
( ( inv g `  g ) `  x
) ( +g  `  g
) y )  e.  s ) } )
21relmpt2opab 6243 . 2  |-  Rel  ( G ~QG  S )
3 releqg.r . . 3  |-  R  =  ( G ~QG  S )
43releqi 4809 . 2  |-  ( Rel 
R  <->  Rel  ( G ~QG  S ) )
52, 4mpbir 200 1  |-  Rel  R
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1633    e. wcel 1701   _Vcvv 2822    C_ wss 3186   {cpr 3675   Rel wrel 4731   ` cfv 5292  (class class class)co 5900   Basecbs 13195   +g cplusg 13255   inv gcminusg 14412   ~QG cqg 14666
This theorem is referenced by:  eqger  14716  eqgid  14718  tgptsmscls  17884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-eqg 14669
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