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Theorem mptv 4112
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 4079 . 2  |-  ( x  e.  _V  |->  B )  =  { <. x ,  y >.  |  ( x  e.  _V  /\  y  =  B ) }
2 vex 2791 . . . 4  |-  x  e. 
_V
32biantrur 492 . . 3  |-  ( y  =  B  <->  ( x  e.  _V  /\  y  =  B ) )
43opabbii 4083 . 2  |-  { <. x ,  y >.  |  y  =  B }  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  y  =  B ) }
51, 4eqtr4i 2306 1  |-  ( x  e.  _V  |->  B )  =  { <. x ,  y >.  |  y  =  B }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {copab 4076    e. cmpt 4077
This theorem is referenced by:  df1st2  6205  df2nd2  6206  fsplit  6223  rankf  7466  cnmptid  17355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790  df-opab 4078  df-mpt 4079
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