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Theorem isoid 5704
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 5398 . 2  |-  (  _I  |`  A ) : A -1-1-onto-> A
2 fvresi 5606 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
3 fvresi 5606 . . . . 5  |-  ( y  e.  A  ->  (
(  _I  |`  A ) `
 y )  =  y )
42, 3breqan12d 3940 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( (  _I  |`  A ) `  x
) R ( (  _I  |`  A ) `  y )  <->  x R
y ) )
54bicomd 140 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <-> 
( (  _I  |`  A ) `
 x ) R ( (  _I  |`  A ) `
 y ) ) )
65rgen2a 2484 . 2  |-  A. x  e.  A  A. y  e.  A  ( x R y  <->  ( (  _I  |`  A ) `  x ) R ( (  _I  |`  A ) `
 y ) )
7 df-isom 5127 . 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 926 1  |-  (  _I  |`  A )  Isom  R ,  R  ( A ,  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 1480   A.wral 2414   class class class wbr 3924    _I cid 4205    |` cres 4536   -1-1-onto->wf1o 5117   ` cfv 5118    Isom wiso 5119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127
This theorem is referenced by:  ordiso  6914
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