Theorem releqg 19089

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.

Step	Hyp	Ref	Expression
1		df-eqg 19039	. . 3 ⊢ ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠)})
2	1	relmpoopab 8050	. 2 ⊢ Rel (𝐺 ~QG 𝑆)
3		releqg.r	. . 3 ⊢ 𝑅 = (𝐺 ~QG 𝑆)
4	3	releqi 5732	. 2 ⊢ (Rel 𝑅 ↔ Rel (𝐺 ~QG 𝑆))
5	2, 4	mpbir 231	1 ⊢ Rel 𝑅

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ⊆ wss 3911  {cpr 4587  Rel wrel 5636  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  +_gcplusg 17196  inv_gcminusg 18848   ~_QG cqg 19036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-eqg 19039

This theorem is referenced by:  eqger  19092  eqgid  19094  tgptsmscls  24070