Theorem mptresid 4997

Description: The restricted identity relation expressed in maps-to notation. (Contributed by FL, 25-Apr-2012.)

Assertion

Ref	Expression
mptresid	|- ( _I |` A ) = ( x e. A |-> x )

Distinct variable group:   x, A

Proof of Theorem mptresid

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opabresid 4996	. 2 |- ( _I |` A ) = { <. x , y >. | ( x e. A /\ y = x ) }
2		df-mpt 4093	. 2 |- ( x e. A |-> x ) = { <. x , y >. | ( x e. A /\ y = x ) }
3	1, 2	eqtr4i 2217	1 |- ( _I |` A ) = ( x e. A |-> x )

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2164   {copab 4090   |-> cmpt 4091   _I cid 4320   |` cres 4662

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-res 4672

This theorem is referenced by:  idref  5800  restid2  12862  txswaphmeolem  14499  dvexp  14890  dvmptidcn  14893  dvmptid  14895  plyid  14925