Theorem opabresid 5091

Description: The restricted identity relation expressed as an ordered-pair class abstraction. (Contributed by FL, 25-Apr-2012.)

Assertion

Ref	Expression
opabresid	|- ( _I |` A ) = { <. x , y >. | ( x e. A /\ y = x ) }

Distinct variable group:   x, A, y

Proof of Theorem opabresid

Step	Hyp	Ref	Expression
1		df-id 4414	. . . 4 |- _I = { <. x , y >. | x = y }
2		equcom 1754	. . . . 5 |- ( x = y <-> y = x )
3	2	opabbii 4177	. . . 4 |- { <. x , y >. | x = y } = { <. x , y >. | y = x }
4	1, 3	eqtri 2253	. . 3 |- _I = { <. x , y >. | y = x }
5	4	reseq1i 5034	. 2 |- ( _I |` A ) = ( { <. x , y >. | y = x } |` A )
6		resopab 5082	. 2 |- ( { <. x , y >. | y = x } |` A ) = { <. x , y >. | ( x e. A /\ y = x ) }
7	5, 6	eqtri 2253	1 |- ( _I |` A ) = { <. x , y >. | ( x e. A /\ y = x ) }

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2203   {copab 4170   _I cid 4409   |` cres 4751

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-res 4761

This theorem is referenced by:  mptresid  5092