Theorem mptv 4025

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 3991	. 2 |- ( x e. _V |-> B ) = { <. x , y >. | ( x e. _V /\ y = B ) }
2		vex 2689	. . . 4 |- x e. _V
3	2	biantrur 301	. . 3 |- ( y = B <-> ( x e. _V /\ y = B ) )
4	3	opabbii 3995	. 2 |- { <. x , y >. | y = B } = { <. x , y >. | ( x e. _V /\ y = B ) }
5	1, 4	eqtr4i 2163	1 |- ( x e. _V |-> B ) = { <. x , y >. | y = B }

Syntax hints:   /\ wa 103   = wceq 1331   e. wcel 1480   _Vcvv 2686   {copab 3988   |-> cmpt 3989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688  df-opab 3990  df-mpt 3991

This theorem is referenced by:  df1st2  6116  df2nd2  6117  hashennn  10526  cnmptid  12450