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Theorem mpt2v 5937
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 5863 . 2  |-  ( x  e.  _V ,  y  e.  _V  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  _V  /\  y  e.  _V )  /\  z  =  C
) }
2 vex 2791 . . . . 5  |-  x  e. 
_V
3 vex 2791 . . . . 5  |-  y  e. 
_V
42, 3pm3.2i 441 . . . 4  |-  ( x  e.  _V  /\  y  e.  _V )
54biantrur 492 . . 3  |-  ( z  =  C  <->  ( (
x  e.  _V  /\  y  e.  _V )  /\  z  =  C
) )
65oprabbii 5903 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  C }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
_V  /\  y  e.  _V )  /\  z  =  C ) }
71, 6eqtr4i 2306 1  |-  ( x  e.  _V ,  y  e.  _V  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  z  =  C }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {coprab 5859    e. cmpt2 5860
This theorem is referenced by:  1st2val  6145  2nd2val  6146
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-oprab 5862  df-mpt2 5863
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