Theorem iunid 3876

Description: An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)

Assertion

Ref	Expression
iunid	|- U_ x e. A { x } = A

Distinct variable group:   x, A

Proof of Theorem iunid

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sn 3538	. . . . 5 |- { x } = { y | y = x }
2		equcom 1683	. . . . . 6 |- ( y = x <-> x = y )
3	2	abbii 2256	. . . . 5 |- { y | y = x } = { y | x = y }
4	1, 3	eqtri 2161	. . . 4 |- { x } = { y | x = y }
5	4	a1i 9	. . 3 |- ( x e. A -> { x } = { y | x = y } )
6	5	iuneq2i 3839	. 2 |- U_ x e. A { x } = U_ x e. A { y | x = y }
7		iunab 3867	. . 3 |- U_ x e. A { y | x = y } = { y | E. x e. A x = y }
8		risset 2466	. . . 4 |- ( y e. A <-> E. x e. A x = y )
9	8	abbii 2256	. . 3 |- { y | y e. A } = { y | E. x e. A x = y }
10		abid2 2261	. . 3 |- { y | y e. A } = A
11	7, 9, 10	3eqtr2i 2167	. 2 |- U_ x e. A { y | x = y } = A
12	6, 11	eqtri 2161	1 |- U_ x e. A { x } = A

Syntax hints:   = wceq 1332   e. wcel 1481   {cab 2126   E.wrex 2418   {csn 3532   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-in 3082  df-ss 3089  df-sn 3538  df-iun 3823

This theorem is referenced by:  abnexg  4375  iunxpconst  4607  xpexgALT  6039  uniqs  6495