Theorem iunid 3791

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 3456	. . . . 5 |- { x } = { y | y = x }
2		equcom 1640	. . . . . 6 |- ( y = x <-> x = y )
3	2	abbii 2204	. . . . 5 |- { y | y = x } = { y | x = y }
4	1, 3	eqtri 2109	. . . 4 |- { x } = { y | x = y }
5	4	a1i 9	. . 3 |- ( x e. A -> { x } = { y | x = y } )
6	5	iuneq2i 3754	. 2 |- U_ x e. A { x } = U_ x e. A { y | x = y }
7		iunab 3782	. . 3 |- U_ x e. A { y | x = y } = { y | E. x e. A x = y }
8		risset 2407	. . . 4 |- ( y e. A <-> E. x e. A x = y )
9	8	abbii 2204	. . 3 |- { y | y e. A } = { y | E. x e. A x = y }
10		abid2 2209	. . 3 |- { y | y e. A } = A
11	7, 9, 10	3eqtr2i 2115	. 2 |- U_ x e. A { y | x = y } = A
12	6, 11	eqtri 2109	1 |- U_ x e. A { x } = A

Syntax hints:   = wceq 1290   e. wcel 1439   {cab 2075   E.wrex 2361   {csn 3450   U_ciun 3736

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-in 3006  df-ss 3013  df-sn 3456  df-iun 3738

This theorem is referenced by:  abnexg  4281  iunxpconst  4511  xpexgALT  5918  uniqs  6364