Metamath Proof Explorer


Theorem dfiun2

Description: Alternate definition of indexed union when B is a set. Definition 15(a) of Suppes p. 44. (Contributed by NM, 27-Jun-1998) (Revised by David Abernethy, 19-Jun-2012)

Ref Expression
Hypothesis dfiun2.1
|- B e. _V
Assertion dfiun2
|- U_ x e. A B = U. { y | E. x e. A y = B }

Proof

Step Hyp Ref Expression
1 dfiun2.1
 |-  B e. _V
2 dfiun2g
 |-  ( A. x e. A B e. _V -> U_ x e. A B = U. { y | E. x e. A y = B } )
3 1 a1i
 |-  ( x e. A -> B e. _V )
4 2 3 mprg
 |-  U_ x e. A B = U. { y | E. x e. A y = B }