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Theorem mpov 5943
Description: Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
Assertion
Ref Expression
mpov  |-  ( x  e.  _V ,  y  e.  _V  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  z  =  C }
Distinct variable groups:    x, z    y,
z    z, C
Allowed substitution hints:    C( x, y)

Proof of Theorem mpov
StepHypRef Expression
1 df-mpo 5858 . 2  |-  ( x  e.  _V ,  y  e.  _V  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  _V  /\  y  e.  _V )  /\  z  =  C
) }
2 vex 2733 . . . . 5  |-  x  e. 
_V
3 vex 2733 . . . . 5  |-  y  e. 
_V
42, 3pm3.2i 270 . . . 4  |-  ( x  e.  _V  /\  y  e.  _V )
54biantrur 301 . . 3  |-  ( z  =  C  <->  ( (
x  e.  _V  /\  y  e.  _V )  /\  z  =  C
) )
65oprabbii 5908 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  C }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
_V  /\  y  e.  _V )  /\  z  =  C ) }
71, 6eqtr4i 2194 1  |-  ( x  e.  _V ,  y  e.  _V  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  z  =  C }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1348    e. wcel 2141   _Vcvv 2730   {coprab 5854    e. cmpo 5855
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732  df-oprab 5857  df-mpo 5858
This theorem is referenced by: (None)
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