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Theorem mptv 4191
Description: Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
Assertion
Ref Expression
mptv  |-  ( x  e.  _V  |->  B )  =  { <. x ,  y >.  |  y  =  B }
Distinct variable groups:    x, y    y, B
Allowed substitution hint:    B( x)

Proof of Theorem mptv
StepHypRef Expression
1 df-mpt 4158 . 2  |-  ( x  e.  _V  |->  B )  =  { <. x ,  y >.  |  ( x  e.  _V  /\  y  =  B ) }
2 vex 2867 . . . 4  |-  x  e. 
_V
32biantrur 492 . . 3  |-  ( y  =  B  <->  ( x  e.  _V  /\  y  =  B ) )
43opabbii 4162 . 2  |-  { <. x ,  y >.  |  y  =  B }  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  y  =  B ) }
51, 4eqtr4i 2381 1  |-  ( x  e.  _V  |->  B )  =  { <. x ,  y >.  |  y  =  B }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864   {copab 4155    e. cmpt 4156
This theorem is referenced by:  df1st2  6289  df2nd2  6290  fsplit  6307  rankf  7553  cnmptid  17455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-v 2866  df-opab 4157  df-mpt 4158
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