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Theorem isoid 5842
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 5527 . 2  |-  (  _I  |`  A ) : A -1-1-onto-> A
2 fvresi 5727 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
3 fvresi 5727 . . . . 5  |-  ( y  e.  A  ->  (
(  _I  |`  A ) `
 y )  =  y )
42, 3breqan12d 4054 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( (  _I  |`  A ) `  x
) R ( (  _I  |`  A ) `  y )  <->  x R
y ) )
54bicomd 192 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <-> 
( (  _I  |`  A ) `
 x ) R ( (  _I  |`  A ) `
 y ) ) )
65rgen2a 2622 . 2  |-  A. x  e.  A  A. y  e.  A  ( x R y  <->  ( (  _I  |`  A ) `  x ) R ( (  _I  |`  A ) `
 y ) )
7 df-isom 5280 . 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 886 1  |-  (  _I  |`  A )  Isom  R ,  R  ( A ,  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1696   A.wral 2556   class class class wbr 4039    _I cid 4320    |` cres 4707   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272
This theorem is referenced by:  ordiso  7247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280
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