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Theorem isexid 20984
Description: The predicate  G has a left and right identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
isexid.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
isexid  |-  ( G  e.  A  ->  ( G  e.  ExId  <->  E. x  e.  X  A. y  e.  X  ( (
x G y )  =  y  /\  (
y G x )  =  y ) ) )
Distinct variable groups:    x, G, y    x, X, y
Allowed substitution hints:    A( x, y)

Proof of Theorem isexid
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 dmeq 4879 . . . . 5  |-  ( g  =  G  ->  dom  g  =  dom  G )
21dmeqd 4881 . . . 4  |-  ( g  =  G  ->  dom  dom  g  =  dom  dom  G )
3 isexid.1 . . . 4  |-  X  =  dom  dom  G
42, 3syl6eqr 2333 . . 3  |-  ( g  =  G  ->  dom  dom  g  =  X )
5 oveq 5864 . . . . . 6  |-  ( g  =  G  ->  (
x g y )  =  ( x G y ) )
65eqeq1d 2291 . . . . 5  |-  ( g  =  G  ->  (
( x g y )  =  y  <->  ( x G y )  =  y ) )
7 oveq 5864 . . . . . 6  |-  ( g  =  G  ->  (
y g x )  =  ( y G x ) )
87eqeq1d 2291 . . . . 5  |-  ( g  =  G  ->  (
( y g x )  =  y  <->  ( y G x )  =  y ) )
96, 8anbi12d 691 . . . 4  |-  ( g  =  G  ->  (
( ( x g y )  =  y  /\  ( y g x )  =  y )  <->  ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) )
104, 9raleqbidv 2748 . . 3  |-  ( g  =  G  ->  ( A. y  e.  dom  dom  g ( ( x g y )  =  y  /\  ( y g x )  =  y )  <->  A. y  e.  X  ( (
x G y )  =  y  /\  (
y G x )  =  y ) ) )
114, 10rexeqbidv 2749 . 2  |-  ( g  =  G  ->  ( E. x  e.  dom  dom  g A. y  e. 
dom  dom  g ( ( x g y )  =  y  /\  (
y g x )  =  y )  <->  E. x  e.  X  A. y  e.  X  ( (
x G y )  =  y  /\  (
y G x )  =  y ) ) )
12 df-exid 20982 . 2  |-  ExId  =  { g  |  E. x  e.  dom  dom  g A. y  e.  dom  dom  g ( ( x g y )  =  y  /\  ( y g x )  =  y ) }
1311, 12elab2g 2916 1  |-  ( G  e.  A  ->  ( G  e.  ExId  <->  E. x  e.  X  A. y  e.  X  ( (
x G y )  =  y  /\  (
y G x )  =  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   dom cdm 4689  (class class class)co 5858    ExId cexid 20981
This theorem is referenced by:  opidon  20989  isexid2  20992  ismndo  21010  exidres  26568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-dm 4699  df-iota 5219  df-fv 5263  df-ov 5861  df-exid 20982
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