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Theorem dfid4 27519
Description: The identity function using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017.)
Assertion
Ref Expression
dfid4  |-  _I  =  ( x  e.  _V  |->  x )

Proof of Theorem dfid4
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 equcom 1734 . . . 4  |-  ( x  =  y  <->  y  =  x )
2 vex 3074 . . . . 5  |-  x  e. 
_V
32biantrur 506 . . . 4  |-  ( y  =  x  <->  ( x  e.  _V  /\  y  =  x ) )
41, 3bitri 249 . . 3  |-  ( x  =  y  <->  ( x  e.  _V  /\  y  =  x ) )
54opabbii 4457 . 2  |-  { <. x ,  y >.  |  x  =  y }  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  y  =  x ) }
6 df-id 4737 . 2  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
7 df-mpt 4453 . 2  |-  ( x  e.  _V  |->  x )  =  { <. x ,  y >.  |  ( x  e.  _V  /\  y  =  x ) }
85, 6, 73eqtr4i 2490 1  |-  _I  =  ( x  e.  _V  |->  x )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3071   {copab 4450    |-> cmpt 4451    _I cid 4732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3073  df-opab 4452  df-mpt 4453  df-id 4737
This theorem is referenced by: (None)
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