Theorem opabresid 4999

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 4328	. . . 4 |- _I = { <. x , y >. | x = y }
2		equcom 1720	. . . . 5 |- ( x = y <-> y = x )
3	2	opabbii 4100	. . . 4 |- { <. x , y >. | x = y } = { <. x , y >. | y = x }
4	1, 3	eqtri 2217	. . 3 |- _I = { <. x , y >. | y = x }
5	4	reseq1i 4942	. 2 |- ( _I |` A ) = ( { <. x , y >. | y = x } |` A )
6		resopab 4990	. 2 |- ( { <. x , y >. | y = x } |` A ) = { <. x , y >. | ( x e. A /\ y = x ) }
7	5, 6	eqtri 2217	1 |- ( _I |` A ) = { <. x , y >. | ( x e. A /\ y = x ) }

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2167   {copab 4093   _I cid 4323   |` cres 4665

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-res 4675

This theorem is referenced by:  mptresid  5000