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Theorem mptv 4476
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 4444 . 2  |-  ( x  e.  _V  |->  B )  =  { <. x ,  y >.  |  ( x  e.  _V  /\  y  =  B ) }
2 vex 3054 . . . 4  |-  x  e. 
_V
32biantrur 504 . . 3  |-  ( y  =  B  <->  ( x  e.  _V  /\  y  =  B ) )
43opabbii 4448 . 2  |-  { <. x ,  y >.  |  y  =  B }  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  y  =  B ) }
51, 4eqtr4i 2428 1  |-  ( x  e.  _V  |->  B )  =  { <. x ,  y >.  |  y  =  B }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1399    e. wcel 1836   _Vcvv 3051   {copab 4441    |-> cmpt 4442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-v 3053  df-opab 4443  df-mpt 4444
This theorem is referenced by:  df1st2  6807  df2nd2  6808  fsplit  6826  rankf  8147  cnmptid  20270
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