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Theorem releqg 14664
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 14620 . . 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 6201 . 2  |-  Rel  ( G ~QG  S )
3 releqg.r . . 3  |-  R  =  ( G ~QG  S )
43releqi 4772 . 2  |-  ( Rel 
R  <->  Rel  ( G ~QG  S ) )
52, 4mpbir 200 1  |-  Rel  R
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {cpr 3641   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   inv gcminusg 14363   ~QG cqg 14617
This theorem is referenced by:  eqger  14667  eqgid  14669  tgptsmscls  17832
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-eqg 14620
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