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Theorem mptv 4061
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 4027 . 2  |-  ( x  e.  _V  |->  B )  =  { <. x ,  y >.  |  ( x  e.  _V  /\  y  =  B ) }
2 vex 2715 . . . 4  |-  x  e. 
_V
32biantrur 301 . . 3  |-  ( y  =  B  <->  ( x  e.  _V  /\  y  =  B ) )
43opabbii 4031 . 2  |-  { <. x ,  y >.  |  y  =  B }  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  y  =  B ) }
51, 4eqtr4i 2181 1  |-  ( x  e.  _V  |->  B )  =  { <. x ,  y >.  |  y  =  B }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1335    e. wcel 2128   _Vcvv 2712   {copab 4024    |-> cmpt 4025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714  df-opab 4026  df-mpt 4027
This theorem is referenced by:  df1st2  6163  df2nd2  6164  hashennn  10649  cnmptid  12668
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