Theorem iunid 3957

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 3613	. . . . 5 |- { x } = { y | y = x }
2		equcom 1717	. . . . . 6 |- ( y = x <-> x = y )
3	2	abbii 2305	. . . . 5 |- { y | y = x } = { y | x = y }
4	1, 3	eqtri 2210	. . . 4 |- { x } = { y | x = y }
5	4	a1i 9	. . 3 |- ( x e. A -> { x } = { y | x = y } )
6	5	iuneq2i 3919	. 2 |- U_ x e. A { x } = U_ x e. A { y | x = y }
7		iunab 3948	. . 3 |- U_ x e. A { y | x = y } = { y | E. x e. A x = y }
8		risset 2518	. . . 4 |- ( y e. A <-> E. x e. A x = y )
9	8	abbii 2305	. . 3 |- { y | y e. A } = { y | E. x e. A x = y }
10		abid2 2310	. . 3 |- { y | y e. A } = A
11	7, 9, 10	3eqtr2i 2216	. 2 |- U_ x e. A { y | x = y } = A
12	6, 11	eqtri 2210	1 |- U_ x e. A { x } = A

Syntax hints:   = wceq 1364   e. wcel 2160   {cab 2175   E.wrex 2469   {csn 3607   U_ciun 3901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-sn 3613  df-iun 3903

This theorem is referenced by:  abnexg  4464  iunxpconst  4704  xpexgALT  6159  uniqs  6620