Theorem mptv 4493

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

Step	Hyp	Ref	Expression
1		df-mpt 4461	. 2 |- ( x e. _V |-> B ) = { <. x , y >. | ( x e. _V /\ y = B ) }
2		vex 3081	. . . 4 |- x e. _V
3	2	biantrur 506	. . 3 |- ( y = B <-> ( x e. _V /\ y = B ) )
4	3	opabbii 4465	. 2 |- { <. x , y >. | y = B } = { <. x , y >. | ( x e. _V /\ y = B ) }
5	1, 4	eqtr4i 2486	1 |- ( x e. _V |-> B ) = { <. x , y >. | y = B }

Syntax hints:   /\ wa 369   = wceq 1370   e. wcel 1758   _Vcvv 3078   {copab 4458   |-> cmpt 4459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3080  df-opab 4460  df-mpt 4461

This theorem is referenced by:  df1st2  6770  df2nd2  6771  fsplit  6788  rankf  8113  cnmptid  19367