Theorem isoid 5989

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 5659	. 2 |- ( _I |` A ) : A -1-1-onto-> A
2		fvresi 5882	. . . . 5 |- ( x e. A -> ( ( _I |` A ) ` x ) = x )
3		fvresi 5882	. . . . 5 |- ( y e. A -> ( ( _I |` A ) ` y ) = y )
4	2, 3	breqan12d 4130	. . . 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 2598	. 2 |- A. x e. A A. y e. A ( x R y <-> ( ( _I |` A ) ` x ) R ( ( _I |` A ) ` y ) )
7		df-isom 5366	. 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 951	1 |- ( _I |` A ) Isom R , R ( A , A )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2205   A.wral 2522   class class class wbr 4114   _I cid 4414   |` cres 4756   -1-1-onto->wf1o 5356   ` cfv 5357   Isom wiso 5358

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366

This theorem is referenced by:  ordiso  7340