Theorem isoid 6579

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 6174	. 2 |- ( _I |` A ) : A -1-1-onto-> A
2		fvresi 6439	. . . . 5 |- ( x e. A -> ( ( _I |` A ) ` x ) = x )
3		fvresi 6439	. . . . 5 |- ( y e. A -> ( ( _I |` A ) ` y ) = y )
4	2, 3	breqan12d 4669	. . . 4 |- ( ( x e. A /\ y e. A ) -> ( ( ( _I |` A ) ` x ) R ( ( _I |` A ) ` y ) <-> x R y ) )
5	4	bicomd 213	. . 3 |- ( ( x e. A /\ y e. A ) -> ( x R y <-> ( ( _I |` A ) ` x ) R ( ( _I |` A ) ` y ) ) )
6	5	rgen2a 2977	. 2 |- A. x e. A A. y e. A ( x R y <-> ( ( _I |` A ) ` x ) R ( ( _I |` A ) ` y ) )
7		df-isom 5897	. 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 955	1 |- ( _I |` A ) Isom R , R ( A , A )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   class class class wbr 4653   _I cid 5023   |` cres 5116   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897

This theorem is referenced by:  ordiso  8421