Theorem dfiun2 3855

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.)

Hypothesis

Ref	Expression
dfiun2.1	|- B e. _V

Assertion

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

Distinct variable groups:   x, y   y, A   y, B

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem dfiun2

Step	Hyp	Ref	Expression
1		dfiun2g 3853	. 2 |- ( A. x e. A B e. _V -> U_ x e. A B = U. { y | E. x e. A y = B } )
2		dfiun2.1	. . 3 |- B e. _V
3	2	a1i 9	. 2 |- ( x e. A -> B e. _V )
4	1, 3	mprg 2492	1 |- U_ x e. A B = U. { y | E. x e. A y = B }

Syntax hints:   = wceq 1332   e. wcel 1481   {cab 2126   E.wrex 2418   _Vcvv 2689   U.cuni 3744   U_ciun 3821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-uni 3745  df-iun 3823

This theorem is referenced by:  funcnvuni  5200  fun11iun  5396  tfrlem8  6223