Theorem mptresid 5012

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 5011	. 2 |- ( _I |` A ) = { <. x , y >. | ( x e. A /\ y = x ) }
2		df-mpt 4106	. 2 |- ( x e. A |-> x ) = { <. x , y >. | ( x e. A /\ y = x ) }
3	1, 2	eqtr4i 2228	1 |- ( _I |` A ) = ( x e. A |-> x )

Syntax hints:   /\ wa 104   = wceq 1372   e. wcel 2175   {copab 4103   |-> cmpt 4104   _I cid 4334   |` cres 4676

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-res 4686

This theorem is referenced by:  idref  5824  restid2  13022  txswaphmeolem  14734  dvexp  15125  dvmptidcn  15128  dvmptid  15130  plyid  15160