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Theorem releqg 17464
 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 17416 . . 3 ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠)})
21relmpt2opab 7146 . 2 Rel (𝐺 ~QG 𝑆)
3 releqg.r . . 3 𝑅 = (𝐺 ~QG 𝑆)
43releqi 5125 . 2 (Rel 𝑅 ↔ Rel (𝐺 ~QG 𝑆))
52, 4mpbir 220 1 Rel 𝑅
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  {cpr 4127  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  invgcminusg 17246   ~QG cqg 17413 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-eqg 17416 This theorem is referenced by:  eqger  17467  eqgid  17469  tgptsmscls  21763
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