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Theorem dfid4 29344
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 1799 . . . 4  |-  ( x  =  y  <->  y  =  x )
2 vex 3109 . . . . 5  |-  x  e. 
_V
32biantrur 504 . . . 4  |-  ( y  =  x  <->  ( x  e.  _V  /\  y  =  x ) )
41, 3bitri 249 . . 3  |-  ( x  =  y  <->  ( x  e.  _V  /\  y  =  x ) )
54opabbii 4503 . 2  |-  { <. x ,  y >.  |  x  =  y }  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  y  =  x ) }
6 df-id 4784 . 2  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
7 df-mpt 4499 . 2  |-  ( x  e.  _V  |->  x )  =  { <. x ,  y >.  |  ( x  e.  _V  /\  y  =  x ) }
85, 6, 73eqtr4i 2493 1  |-  _I  =  ( x  e.  _V  |->  x )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   {copab 4496    |-> cmpt 4497    _I cid 4779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3108  df-opab 4498  df-mpt 4499  df-id 4784
This theorem is referenced by:  dfid5  38195  dfid6  38196
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