Theorem mpov 5943

Description: Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)

Assertion

Ref	Expression
mpov	|- ( x e. _V , y e. _V |-> C ) = { <. <. x , y >. , z >. | z = C }

Distinct variable groups:   x, z   y, z   z, C

Allowed substitution hints:   C( x, y)

Proof of Theorem mpov

Step	Hyp	Ref	Expression
1		df-mpo 5858	. 2 |- ( x e. _V , y e. _V |-> C ) = { <. <. x , y >. , z >. | ( ( x e. _V /\ y e. _V ) /\ z = C ) }
2		vex 2733	. . . . 5 |- x e. _V
3		vex 2733	. . . . 5 |- y e. _V
4	2, 3	pm3.2i 270	. . . 4 |- ( x e. _V /\ y e. _V )
5	4	biantrur 301	. . 3 |- ( z = C <-> ( ( x e. _V /\ y e. _V ) /\ z = C ) )
6	5	oprabbii 5908	. 2 |- { <. <. x , y >. , z >. | z = C } = { <. <. x , y >. , z >. | ( ( x e. _V /\ y e. _V ) /\ z = C ) }
7	1, 6	eqtr4i 2194	1 |- ( x e. _V , y e. _V |-> C ) = { <. <. x , y >. , z >. | z = C }

Syntax hints:   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   {coprab 5854   e. cmpo 5855

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-oprab 5857  df-mpo 5858

This theorem is referenced by: (None)