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Theorem mptv 4332
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 4299 . 2  |-  ( x  e.  _V  |->  B )  =  { <. x ,  y >.  |  ( x  e.  _V  /\  y  =  B ) }
2 vex 2968 . . . 4  |-  x  e. 
_V
32biantrur 494 . . 3  |-  ( y  =  B  <->  ( x  e.  _V  /\  y  =  B ) )
43opabbii 4303 . 2  |-  { <. x ,  y >.  |  y  =  B }  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  y  =  B ) }
51, 4eqtr4i 2466 1  |-  ( x  e.  _V  |->  B )  =  { <. x ,  y >.  |  y  =  B }
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1654    e. wcel 1728   _Vcvv 2965   {copab 4296    e. cmpt 4297
This theorem is referenced by:  df1st2  6469  df2nd2  6470  fsplit  6487  rankf  7756  cnmptid  17731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-v 2967  df-opab 4298  df-mpt 4299
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