Theorem mptv 4086

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 4052	. 2 |- ( x e. _V |-> B ) = { <. x , y >. | ( x e. _V /\ y = B ) }
2		vex 2733	. . . 4 |- x e. _V
3	2	biantrur 301	. . 3 |- ( y = B <-> ( x e. _V /\ y = B ) )
4	3	opabbii 4056	. 2 |- { <. x , y >. | y = B } = { <. x , y >. | ( x e. _V /\ y = B ) }
5	1, 4	eqtr4i 2194	1 |- ( x e. _V |-> B ) = { <. x , y >. | y = B }

Syntax hints:   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   {copab 4049   |-> cmpt 4050

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732  df-opab 4051  df-mpt 4052

This theorem is referenced by:  df1st2  6198  df2nd2  6199  hashennn  10714  cnmptid  13075