Theorem relrngo 33695

Description: The class of all unital rings is a relation. (Contributed by FL, 31-Aug-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
relrngo	|- Rel RingOps

Proof of Theorem relrngo

Dummy variables g h x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rngo 33694	. 2 |- RingOps = { <. g , h >. | ( ( g e. AbelOp /\ h : ( ran g X. ran g ) --> ran g ) /\ ( A. x e. ran g A. y e. ran g A. z e. ran g ( ( ( x h y ) h z ) = ( x h ( y h z ) ) /\ ( x h ( y g z ) ) = ( ( x h y ) g ( x h z ) ) /\ ( ( x g y ) h z ) = ( ( x h z ) g ( y h z ) ) ) /\ E. x e. ran g A. y e. ran g ( ( x h y ) = y /\ ( y h x ) = y ) ) ) }
2	1	relopabi 5245	1 |- Rel RingOps

Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   X. cxp 5112   ran crn 5115   Rel wrel 5119   -->wf 5884  (class class class)co 6650   AbelOpcablo 27398   RingOpscrngo 33693

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  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-opab 4713  df-xp 5120  df-rel 5121  df-rngo 33694

This theorem is referenced by:  isrngo  33696  rngoi  33698  rngoablo2  33708  rngosn3  33723  isdrngo1  33755  iscrngo2  33796