Theorem mpov 6008

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 5923	. 2 |- ( x e. _V , y e. _V |-> C ) = { <. <. x , y >. , z >. | ( ( x e. _V /\ y e. _V ) /\ z = C ) }
2		vex 2763	. . . . 5 |- x e. _V
3		vex 2763	. . . . 5 |- y e. _V
4	2, 3	pm3.2i 272	. . . 4 |- ( x e. _V /\ y e. _V )
5	4	biantrur 303	. . 3 |- ( z = C <-> ( ( x e. _V /\ y e. _V ) /\ z = C ) )
6	5	oprabbii 5973	. 2 |- { <. <. x , y >. , z >. | z = C } = { <. <. x , y >. , z >. | ( ( x e. _V /\ y e. _V ) /\ z = C ) }
7	1, 6	eqtr4i 2217	1 |- ( x e. _V , y e. _V |-> C ) = { <. <. x , y >. , z >. | z = C }

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   {coprab 5919   e. cmpo 5920

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-v 2762  df-oprab 5922  df-mpo 5923

This theorem is referenced by: (None)