Theorem mptresid 4943

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 4050	. 2 |- ( x e. A |-> x ) = { <. x , y >. | ( x e. A /\ y = x ) }
2		opabresid 4942	. 2 |- { <. x , y >. | ( x e. A /\ y = x ) } = ( _I |` A )
3	1, 2	eqtri 2191	1 |- ( x e. A |-> x ) = ( _I |` A )

Syntax hints:   /\ wa 103   = wceq 1348   e. wcel 2141   {copab 4047   |-> cmpt 4048   _I cid 4271   |` cres 4611

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-res 4621

This theorem is referenced by:  idref  5733  restid2  12575  txswaphmeolem  13073  dvexp  13428  dvmptidcn  13431