Theorem dfiun2 3900

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 3898	. 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 2523	1 |- U_ x e. A B = U. { y | E. x e. A y = B }

Syntax hints:   = wceq 1343   e. wcel 2136   {cab 2151   E.wrex 2445   _Vcvv 2726   U.cuni 3789   U_ciun 3866

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-uni 3790  df-iun 3868

This theorem is referenced by:  funcnvuni  5257  fun11iun  5453  tfrlem8  6286