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Theorem mpt2v 6166
Description: Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
Assertion
Ref Expression
mpt2v  |-  ( 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 mpt2v
StepHypRef Expression
1 df-mpt2 6089 . 2  |-  ( x  e.  _V ,  y  e.  _V  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  _V  /\  y  e.  _V )  /\  z  =  C
) }
2 vex 2961 . . . . 5  |-  x  e. 
_V
3 vex 2961 . . . . 5  |-  y  e. 
_V
42, 3pm3.2i 443 . . . 4  |-  ( x  e.  _V  /\  y  e.  _V )
54biantrur 494 . . 3  |-  ( z  =  C  <->  ( (
x  e.  _V  /\  y  e.  _V )  /\  z  =  C
) )
65oprabbii 6132 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  C }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
_V  /\  y  e.  _V )  /\  z  =  C ) }
71, 6eqtr4i 2461 1  |-  ( x  e.  _V ,  y  e.  _V  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  z  =  C }
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   {coprab 6085    e. cmpt2 6086
This theorem is referenced by:  1st2val  6375  2nd2val  6376
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 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-v 2960  df-oprab 6088  df-mpt2 6089
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