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Theorem isoid 5501
Description: Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
Assertion
Ref Expression
isoid  |-  (  _I  |`  A )  Isom  R ,  R  ( A ,  A )

Proof of Theorem isoid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oi 5215 . 2  |-  (  _I  |`  A ) : A -1-1-onto-> A
2 fvresi 5408 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
3 fvresi 5408 . . . . 5  |-  ( y  e.  A  ->  (
(  _I  |`  A ) `
 y )  =  y )
42, 3breqan12d 3820 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( (  _I  |`  A ) `  x
) R ( (  _I  |`  A ) `  y )  <->  x R
y ) )
54bicomd 139 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <-> 
( (  _I  |`  A ) `
 x ) R ( (  _I  |`  A ) `
 y ) ) )
65rgen2a 2422 . 2  |-  A. x  e.  A  A. y  e.  A  ( x R y  <->  ( (  _I  |`  A ) `  x ) R ( (  _I  |`  A ) `
 y ) )
7 df-isom 4961 . 2  |-  ( (  _I  |`  A )  Isom  R ,  R  ( A ,  A )  <-> 
( (  _I  |`  A ) : A -1-1-onto-> A  /\  A. x  e.  A  A. y  e.  A  ( x R y  <->  ( (  _I  |`  A ) `  x ) R ( (  _I  |`  A ) `
 y ) ) ) )
81, 6, 7mpbir2an 884 1  |-  (  _I  |`  A )  Isom  R ,  R  ( A ,  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    e. wcel 1434   A.wral 2353   class class class wbr 3805    _I cid 4071    |` cres 4393   -1-1-onto->wf1o 4951   ` cfv 4952    Isom wiso 4953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-isom 4961
This theorem is referenced by:  ordiso  6541
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