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Theorem releqg 18265
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 18216 . . 3 ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠)})
21relmpoopab 7778 . 2 Rel (𝐺 ~QG 𝑆)
3 releqg.r . . 3 𝑅 = (𝐺 ~QG 𝑆)
43releqi 5645 . 2 (Rel 𝑅 ↔ Rel (𝐺 ~QG 𝑆))
52, 4mpbir 232 1 Rel 𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  {cpr 4559  Rel wrel 5553  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  invgcminusg 18042   ~QG cqg 18213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-eqg 18216
This theorem is referenced by:  eqger  18268  eqgid  18270  tgptsmscls  22685
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