Theorem mptresid 4868

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

Assertion

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

Distinct variable group:   x, A

Proof of Theorem mptresid

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 3986	. 2 |- ( x e. A |-> x ) = { <. x , y >. | ( x e. A /\ y = x ) }
2		opabresid 4867	. 2 |- { <. x , y >. | ( x e. A /\ y = x ) } = ( _I |` A )
3	1, 2	eqtri 2158	1 |- ( x e. A |-> x ) = ( _I |` A )

Syntax hints:   /\ wa 103   = wceq 1331   e. wcel 1480   {copab 3983   |-> cmpt 3984   _I cid 4205   |` cres 4536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-res 4546

This theorem is referenced by:  idref  5651  restid2  12118  txswaphmeolem  12478  dvexp  12833  dvmptidcn  12836