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Theorem isoid 6051
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 5715 . 2  |-  (  _I  |`  A ) : A -1-1-onto-> A
2 fvresi 5926 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
3 fvresi 5926 . . . . 5  |-  ( y  e.  A  ->  (
(  _I  |`  A ) `
 y )  =  y )
42, 3breqan12d 4229 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( (  _I  |`  A ) `  x
) R ( (  _I  |`  A ) `  y )  <->  x R
y ) )
54bicomd 194 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <-> 
( (  _I  |`  A ) `
 x ) R ( (  _I  |`  A ) `
 y ) ) )
65rgen2a 2774 . 2  |-  A. x  e.  A  A. y  e.  A  ( x R y  <->  ( (  _I  |`  A ) `  x ) R ( (  _I  |`  A ) `
 y ) )
7 df-isom 5465 . 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 888 1  |-  (  _I  |`  A )  Isom  R ,  R  ( A ,  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    e. wcel 1726   A.wral 2707   class class class wbr 4214    _I cid 4495    |` cres 4882   -1-1-onto->wf1o 5455   ` cfv 5456    Isom wiso 5457
This theorem is referenced by:  ordiso  7487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465
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