Theorem releqg 17641

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 17593	. . 3 |- ~QG = ( g e. _V , s e. _V |-> { <. x , y >. | ( { x , y } C_ ( Base ` g ) /\ ( ( ( invg ` g ) ` x ) ( +g ` g ) y ) e. s ) } )
2	1	relmpt2opab 7259	. 2 |- Rel ( G ~QG S )
3		releqg.r	. . 3 |- R = ( G ~QG S )
4	3	releqi 5202	. 2 |- ( Rel R <-> Rel ( G ~QG S ) )
5	2, 4	mpbir 221	1 |- Rel R

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {cpr 4179   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   invgcminusg 17423   ~_QG cqg 17590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-eqg 17593

This theorem is referenced by:  eqger  17644  eqgid  17646  tgptsmscls  21953