Theorem dfiun2 3682

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 3680	. 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 2372	1 |- U_x e. A B = U. {y | E.x e. A y = B }

Syntax hints:   = wceq 1242   e. wcel 1390   {cab 2023  E.wrex 2301   _Vcvv 2551   U.cuni 3571   U_ciun 3648

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-uni 3572  df-iun 3650

This theorem is referenced by:  funcnvuni  4911  fun11iun  5090  tfrlem8  5875