Theorem releqg 19100

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 19050	. . 3 ⊢ ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠)})
2	1	relmpoopab 8077	. 2 ⊢ Rel (𝐺 ~QG 𝑆)
3		releqg.r	. . 3 ⊢ 𝑅 = (𝐺 ~QG 𝑆)
4	3	releqi 5770	. 2 ⊢ (Rel 𝑅 ↔ Rel (𝐺 ~QG 𝑆))
5	2, 4	mpbir 230	1 ⊢ Rel 𝑅

Syntax hints:   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ⊆ wss 3943  {cpr 4625  Rel wrel 5674  ‘cfv 6536  (class class class)co 7404  Basecbs 17151  +_gcplusg 17204  inv_gcminusg 18862   ~_QG cqg 19047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-eqg 19050

This theorem is referenced by:  eqger  19103  eqgid  19105  tgptsmscls  24005