Theorem mpt2v 6750

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 6655	. 2 |- ( x e. _V , y e. _V |-> C ) = { <. <. x , y >. , z >. | ( ( x e. _V /\ y e. _V ) /\ z = C ) }
2		vex 3203	. . . . 5 |- x e. _V
3		vex 3203	. . . . 5 |- y e. _V
4	2, 3	pm3.2i 471	. . . 4 |- ( x e. _V /\ y e. _V )
5	4	biantrur 527	. . 3 |- ( z = C <-> ( ( x e. _V /\ y e. _V ) /\ z = C ) )
6	5	oprabbii 6710	. 2 |- { <. <. x , y >. , z >. | z = C } = { <. <. x , y >. , z >. | ( ( x e. _V /\ y e. _V ) /\ z = C ) }
7	1, 6	eqtr4i 2647	1 |- ( x e. _V , y e. _V |-> C ) = { <. <. x , y >. , z >. | z = C }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {coprab 6651   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  1st2val  7194  2nd2val  7195