Theorem mptresid 5027

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 5026	. 2 |- ( _I |` A ) = { <. x , y >. | ( x e. A /\ y = x ) }
2		df-mpt 4118	. 2 |- ( x e. A |-> x ) = { <. x , y >. | ( x e. A /\ y = x ) }
3	1, 2	eqtr4i 2230	1 |- ( _I |` A ) = ( x e. A |-> x )

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2177   {copab 4115   |-> cmpt 4116   _I cid 4348   |` cres 4690

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-res 4700

This theorem is referenced by:  idref  5843  restid2  13165  txswaphmeolem  14877  dvexp  15268  dvmptidcn  15271  dvmptid  15273  plyid  15303