Theorem dfiun2 3691

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 3689	. 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 2378	1 |- U_ x e. A B = U. { y | E. x e. A y = B }

Syntax hints:   = wceq 1243   e. wcel 1393   {cab 2026   E.wrex 2307   _Vcvv 2557   U.cuni 3580   U_ciun 3657

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-uni 3581  df-iun 3659

This theorem is referenced by:  funcnvuni  4968  fun11iun  5147  tfrlem8  5934