Theorem iunid 3733

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 3404	. . . . 5 |- { x } = { y | y = x }
2		equcom 1633	. . . . . 6 |- ( y = x <-> x = y )
3	2	abbii 2194	. . . . 5 |- { y | y = x } = { y | x = y }
4	1, 3	eqtri 2101	. . . 4 |- { x } = { y | x = y }
5	4	a1i 9	. . 3 |- ( x e. A -> { x } = { y | x = y } )
6	5	iuneq2i 3696	. 2 |- U_ x e. A { x } = U_ x e. A { y | x = y }
7		iunab 3724	. . 3 |- U_ x e. A { y | x = y } = { y | E. x e. A x = y }
8		risset 2394	. . . 4 |- ( y e. A <-> E. x e. A x = y )
9	8	abbii 2194	. . 3 |- { y | y e. A } = { y | E. x e. A x = y }
10		abid2 2199	. . 3 |- { y | y e. A } = A
11	7, 9, 10	3eqtr2i 2107	. 2 |- U_ x e. A { y | x = y } = A
12	6, 11	eqtri 2101	1 |- U_ x e. A { x } = A

Syntax hints:   = wceq 1284   e. wcel 1433   {cab 2067   E.wrex 2349   {csn 3398   U_ciun 3678

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-sn 3404  df-iun 3680

This theorem is referenced by:  iunxpconst  4418  xpexgALT  5780  uniqs  6187