Theorem mpt2v 5614

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

Assertion

Ref	Expression
mpt2v	|- ( 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 mpt2v

Step	Hyp	Ref	Expression
1		df-mpt2 5537	. 2 |- ( x e. _V , y e. _V |-> C ) = { <. <. x , y >. , z >. | ( ( x e. _V /\ y e. _V ) /\ z = C ) }
2		vex 2604	. . . . 5 |- x e. _V
3		vex 2604	. . . . 5 |- y e. _V
4	2, 3	pm3.2i 266	. . . 4 |- ( x e. _V /\ y e. _V )
5	4	biantrur 297	. . 3 |- ( z = C <-> ( ( x e. _V /\ y e. _V ) /\ z = C ) )
6	5	oprabbii 5580	. 2 |- { <. <. x , y >. , z >. | z = C } = { <. <. x , y >. , z >. | ( ( x e. _V /\ y e. _V ) /\ z = C ) }
7	1, 6	eqtr4i 2104	1 |- ( x e. _V , y e. _V |-> C ) = { <. <. x , y >. , z >. | z = C }

Syntax hints:   /\ wa 102   = wceq 1284   e. wcel 1433   _Vcvv 2601   {coprab 5533   |-> cmpt2 5534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603  df-oprab 5536  df-mpt2 5537

This theorem is referenced by: (None)