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Theorem mpt2v 5719
 Description: Operation with universal domain in maps-to notation. (Contributed by set.mm contributors, 16-Aug-2013.)
Assertion
Ref Expression
mpt2v (x V, y 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
StepHypRef Expression
1 df-mpt2 5654 . 2 (x V, y V C) = {x, y, z ((x V y V) z = C)}
2 vex 2862 . . . . 5 x V
3 vex 2862 . . . . 5 y V
42, 3pm3.2i 441 . . . 4 (x V y V)
54biantrur 492 . . 3 (z = C ↔ ((x V y V) z = C))
65oprabbii 5565 . 2 {x, y, z z = C} = {x, y, z ((x V y V) z = C)}
71, 6eqtr4i 2376 1 (x V, y V C) = {x, y, z z = C}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {coprab 5527   ↦ cmpt2 5653 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861  df-oprab 5528  df-mpt2 5654 This theorem is referenced by:  dfswap4  5729
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