Theorem mpt2v 5594

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 5517	. 2 |- ( x e. _V , y e. _V |-> C ) = { <. <. x , y >. , z >. | ( ( x e. _V /\ y e. _V ) /\ z = C ) }
2		vex 2560	. . . . 5 |- x e. _V
3		vex 2560	. . . . 5 |- y e. _V
4	2, 3	pm3.2i 257	. . . 4 |- ( x e. _V /\ y e. _V )
5	4	biantrur 287	. . 3 |- ( z = C <-> ( ( x e. _V /\ y e. _V ) /\ z = C ) )
6	5	oprabbii 5560	. 2 |- { <. <. x , y >. , z >. | z = C } = { <. <. x , y >. , z >. | ( ( x e. _V /\ y e. _V ) /\ z = C ) }
7	1, 6	eqtr4i 2063	1 |- ( x e. _V , y e. _V |-> C ) = { <. <. x , y >. , z >. | z = C }

Syntax hints:   /\ wa 97   = wceq 1243   e. wcel 1393   _Vcvv 2557   {coprab 5513   |-> cmpt2 5514

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559  df-oprab 5516  df-mpt2 5517

This theorem is referenced by: (None)