Theorem mptv 4226

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 4192	. 2 |- ( x e. _V |-> B ) = { <. x , y >. | ( x e. _V /\ y = B ) }
2		vex 2824	. . . 4 |- x e. _V
3	2	biantrur 303	. . 3 |- ( y = B <-> ( x e. _V /\ y = B ) )
4	3	opabbii 4196	. 2 |- { <. x , y >. | y = B } = { <. x , y >. | ( x e. _V /\ y = B ) }
5	1, 4	eqtr4i 2262	1 |- ( x e. _V |-> B ) = { <. x , y >. | y = B }

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   {copab 4189   |-> cmpt 4190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823  df-opab 4191  df-mpt 4192

This theorem is referenced by:  df1st2  6448  df2nd2  6449  hashennn  11200  cnmptid  15308