Theorem mptresid 5065

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 5064	. 2 |- ( _I |` A ) = { <. x , y >. | ( x e. A /\ y = x ) }
2		df-mpt 4150	. 2 |- ( x e. A |-> x ) = { <. x , y >. | ( x e. A /\ y = x ) }
3	1, 2	eqtr4i 2253	1 |- ( _I |` A ) = ( x e. A |-> x )

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   {copab 4147   |-> cmpt 4148   _I cid 4383   |` cres 4725

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-res 4735

This theorem is referenced by:  idref  5892  restid2  13321  txswaphmeolem  15034  dvexp  15425  dvmptidcn  15428  dvmptid  15430  plyid  15460