Theorem iunid 3921

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 3582	. . . . 5 |- { x } = { y | y = x }
2		equcom 1694	. . . . . 6 |- ( y = x <-> x = y )
3	2	abbii 2282	. . . . 5 |- { y | y = x } = { y | x = y }
4	1, 3	eqtri 2186	. . . 4 |- { x } = { y | x = y }
5	4	a1i 9	. . 3 |- ( x e. A -> { x } = { y | x = y } )
6	5	iuneq2i 3884	. 2 |- U_ x e. A { x } = U_ x e. A { y | x = y }
7		iunab 3912	. . 3 |- U_ x e. A { y | x = y } = { y | E. x e. A x = y }
8		risset 2494	. . . 4 |- ( y e. A <-> E. x e. A x = y )
9	8	abbii 2282	. . 3 |- { y | y e. A } = { y | E. x e. A x = y }
10		abid2 2287	. . 3 |- { y | y e. A } = A
11	7, 9, 10	3eqtr2i 2192	. 2 |- U_ x e. A { y | x = y } = A
12	6, 11	eqtri 2186	1 |- U_ x e. A { x } = A

Syntax hints:   = wceq 1343   e. wcel 2136   {cab 2151   E.wrex 2445   {csn 3576   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-in 3122  df-ss 3129  df-sn 3582  df-iun 3868

This theorem is referenced by:  abnexg  4424  iunxpconst  4664  xpexgALT  6101  uniqs  6559