Theorem mptresid 5058

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 5057	. 2 |- ( _I |` A ) = { <. x , y >. | ( x e. A /\ y = x ) }
2		df-mpt 4146	. 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 4143   |-> cmpt 4144   _I cid 4378   |` cres 4720

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 4201  ax-pow 4257  ax-pr 4292

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 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-res 4730

This theorem is referenced by:  idref  5879  restid2  13276  txswaphmeolem  14988  dvexp  15379  dvmptidcn  15382  dvmptid  15384  plyid  15414