Theorem iunid 3863

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 3528	. . . . 5 |- { x } = { y | y = x }
2		equcom 1682	. . . . . 6 |- ( y = x <-> x = y )
3	2	abbii 2253	. . . . 5 |- { y | y = x } = { y | x = y }
4	1, 3	eqtri 2158	. . . 4 |- { x } = { y | x = y }
5	4	a1i 9	. . 3 |- ( x e. A -> { x } = { y | x = y } )
6	5	iuneq2i 3826	. 2 |- U_ x e. A { x } = U_ x e. A { y | x = y }
7		iunab 3854	. . 3 |- U_ x e. A { y | x = y } = { y | E. x e. A x = y }
8		risset 2461	. . . 4 |- ( y e. A <-> E. x e. A x = y )
9	8	abbii 2253	. . 3 |- { y | y e. A } = { y | E. x e. A x = y }
10		abid2 2258	. . 3 |- { y | y e. A } = A
11	7, 9, 10	3eqtr2i 2164	. 2 |- U_ x e. A { y | x = y } = A
12	6, 11	eqtri 2158	1 |- U_ x e. A { x } = A

Syntax hints:   = wceq 1331   e. wcel 1480   {cab 2123   E.wrex 2415   {csn 3522   U_ciun 3808

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-sn 3528  df-iun 3810

This theorem is referenced by:  abnexg  4362  iunxpconst  4594  xpexgALT  6024  uniqs  6480