MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  releqg Structured version   Visualization version   GIF version

Theorem releqg 19130
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 19080 . . 3 ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠)})
21relmpoopab 8099 . 2 Rel (𝐺 ~QG 𝑆)
3 releqg.r . . 3 𝑅 = (𝐺 ~QG 𝑆)
43releqi 5779 . 2 (Rel 𝑅 ↔ Rel (𝐺 ~QG 𝑆))
52, 4mpbir 230 1 Rel 𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1534  wcel 2099  Vcvv 3471  wss 3947  {cpr 4631  Rel wrel 5683  cfv 6548  (class class class)co 7420  Basecbs 17180  +gcplusg 17233  invgcminusg 18891   ~QG cqg 19077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-eqg 19080
This theorem is referenced by:  eqger  19133  eqgid  19135  tgptsmscls  24067
  Copyright terms: Public domain W3C validator