Theorem releqg 19114

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 19064	. . 3 ⊢ ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠)})
2	1	relmpoopab 8076	. 2 ⊢ Rel (𝐺 ~QG 𝑆)
3		releqg.r	. . 3 ⊢ 𝑅 = (𝐺 ~QG 𝑆)
4	3	releqi 5743	. 2 ⊢ (Rel 𝑅 ↔ Rel (𝐺 ~QG 𝑆))
5	2, 4	mpbir 231	1 ⊢ Rel 𝑅

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3917  {cpr 4594  Rel wrel 5646  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  inv_gcminusg 18873   ~_QG cqg 19061

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-eqg 19064

This theorem is referenced by:  eqger  19117  eqgid  19119  tgptsmscls  24044