Theorem isoid 5869

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.

Step	Hyp	Ref	Expression
1		f1oi 5554	. 2 |- ( _I |` A ) : A -1-1-onto-> A
2		fvresi 5767	. . . . 5 |- ( x e. A -> ( ( _I |` A ) ` x ) = x )
3		fvresi 5767	. . . . 5 |- ( y e. A -> ( ( _I |` A ) ` y ) = y )
4	2, 3	breqan12d 4059	. . . 4 |- ( ( x e. A /\ y e. A ) -> ( ( ( _I |` A ) ` x ) R ( ( _I |` A ) ` y ) <-> x R y ) )
5	4	bicomd 141	. . 3 |- ( ( x e. A /\ y e. A ) -> ( x R y <-> ( ( _I |` A ) ` x ) R ( ( _I |` A ) ` y ) ) )
6	5	rgen2a 2559	. 2 |- A. x e. A A. y e. A ( x R y <-> ( ( _I |` A ) ` x ) R ( ( _I |` A ) ` y ) )
7		df-isom 5277	. 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 ) ) ) )
8	1, 6, 7	mpbir2an 944	1 |- ( _I |` A ) Isom R , R ( A , A )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2175   A.wral 2483   class class class wbr 4043   _I cid 4333   |` cres 4675   -1-1-onto->wf1o 5267   ` cfv 5268   Isom wiso 5269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-isom 5277

This theorem is referenced by:  ordiso  7120