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Theorem isexid 23816
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 5052 . . . . 5  |-  ( g  =  G  ->  dom  g  =  dom  G )
21dmeqd 5054 . . . 4  |-  ( g  =  G  ->  dom  dom  g  =  dom  dom  G )
3 isexid.1 . . . 4  |-  X  =  dom  dom  G
42, 3syl6eqr 2493 . . 3  |-  ( g  =  G  ->  dom  dom  g  =  X )
5 oveq 6109 . . . . . 6  |-  ( g  =  G  ->  (
x g y )  =  ( x G y ) )
65eqeq1d 2451 . . . . 5  |-  ( g  =  G  ->  (
( x g y )  =  y  <->  ( x G y )  =  y ) )
7 oveq 6109 . . . . . 6  |-  ( g  =  G  ->  (
y g x )  =  ( y G x ) )
87eqeq1d 2451 . . . . 5  |-  ( g  =  G  ->  (
( y g x )  =  y  <->  ( y G x )  =  y ) )
96, 8anbi12d 710 . . . 4  |-  ( g  =  G  ->  (
( ( x g y )  =  y  /\  ( y g x )  =  y )  <->  ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) )
104, 9raleqbidv 2943 . . 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 2944 . 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 23814 . 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 3120 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 369    = wceq 1369    e. wcel 1756   A.wral 2727   E.wrex 2728   dom cdm 4852  (class class class)co 6103    ExId cexid 23813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-dm 4862  df-iota 5393  df-fv 5438  df-ov 6106  df-exid 23814
This theorem is referenced by:  opidon  23821  isexid2  23824  ismndo  23842  exidres  28755
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