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Theorem isexid 21096
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 4961 . . . . 5  |-  ( g  =  G  ->  dom  g  =  dom  G )
21dmeqd 4963 . . . 4  |-  ( g  =  G  ->  dom  dom  g  =  dom  dom  G )
3 isexid.1 . . . 4  |-  X  =  dom  dom  G
42, 3syl6eqr 2408 . . 3  |-  ( g  =  G  ->  dom  dom  g  =  X )
5 oveq 5951 . . . . . 6  |-  ( g  =  G  ->  (
x g y )  =  ( x G y ) )
65eqeq1d 2366 . . . . 5  |-  ( g  =  G  ->  (
( x g y )  =  y  <->  ( x G y )  =  y ) )
7 oveq 5951 . . . . . 6  |-  ( g  =  G  ->  (
y g x )  =  ( y G x ) )
87eqeq1d 2366 . . . . 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 2824 . . 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 2825 . 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 21094 . 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 2992 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 1642    e. wcel 1710   A.wral 2619   E.wrex 2620   dom cdm 4771  (class class class)co 5945    ExId cexid 21093
This theorem is referenced by:  opidon  21101  isexid2  21104  ismndo  21122  exidres  25891
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-dm 4781  df-iota 5301  df-fv 5345  df-ov 5948  df-exid 21094
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