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Theorem isoid 5786
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 5478 . 2  |-  (  _I  |`  A ) : A -1-1-onto-> A
2 fvresi 5686 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
3 fvresi 5686 . . . . 5  |-  ( y  e.  A  ->  (
(  _I  |`  A ) `
 y )  =  y )
42, 3breqan12d 4003 . . . 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 2524 . 2  |-  A. x  e.  A  A. y  e.  A  ( x R y  <->  ( (  _I  |`  A ) `  x ) R ( (  _I  |`  A ) `
 y ) )
7 df-isom 5205 . 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 937 1  |-  (  _I  |`  A )  Isom  R ,  R  ( A ,  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 2141   A.wral 2448   class class class wbr 3987    _I cid 4271    |` cres 4611   -1-1-onto->wf1o 5195   ` cfv 5196    Isom wiso 5197
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205
This theorem is referenced by:  ordiso  7009
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