Theorem releqg 19055

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 19005	. . 3 ⊢ ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠)})
2	1	relmpoopab 8080	. 2 ⊢ Rel (𝐺 ~QG 𝑆)
3		releqg.r	. . 3 ⊢ 𝑅 = (𝐺 ~QG 𝑆)
4	3	releqi 5778	. 2 ⊢ (Rel 𝑅 ↔ Rel (𝐺 ~QG 𝑆))
5	2, 4	mpbir 230	1 ⊢ Rel 𝑅

Syntax hints:   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ⊆ wss 3949  {cpr 4631  Rel wrel 5682  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  +_gcplusg 17197  inv_gcminusg 18820   ~_QG cqg 19002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-eqg 19005

This theorem is referenced by:  eqger  19058  eqgid  19060  tgptsmscls  23654