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Theorem isexid 25520
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 5192 . . . . 5  |-  ( g  =  G  ->  dom  g  =  dom  G )
21dmeqd 5194 . . . 4  |-  ( g  =  G  ->  dom  dom  g  =  dom  dom  G )
3 isexid.1 . . . 4  |-  X  =  dom  dom  G
42, 3syl6eqr 2513 . . 3  |-  ( g  =  G  ->  dom  dom  g  =  X )
5 oveq 6276 . . . . . 6  |-  ( g  =  G  ->  (
x g y )  =  ( x G y ) )
65eqeq1d 2456 . . . . 5  |-  ( g  =  G  ->  (
( x g y )  =  y  <->  ( x G y )  =  y ) )
7 oveq 6276 . . . . . 6  |-  ( g  =  G  ->  (
y g x )  =  ( y G x ) )
87eqeq1d 2456 . . . . 5  |-  ( g  =  G  ->  (
( y g x )  =  y  <->  ( y G x )  =  y ) )
96, 8anbi12d 708 . . . 4  |-  ( g  =  G  ->  (
( ( x g y )  =  y  /\  ( y g x )  =  y )  <->  ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) )
104, 9raleqbidv 3065 . . 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 3066 . 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 25518 . 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 3245 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   dom cdm 4988  (class class class)co 6270    ExId cexid 25517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-dm 4998  df-iota 5534  df-fv 5578  df-ov 6273  df-exid 25518
This theorem is referenced by:  opidonOLD  25525  isexid2  25528  ismndo  25546  exidres  30583
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