Theorem iunid 4031

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 3679	. . . . 5 |- { x } = { y | y = x }
2		equcom 1754	. . . . . 6 |- ( y = x <-> x = y )
3	2	abbii 2347	. . . . 5 |- { y | y = x } = { y | x = y }
4	1, 3	eqtri 2252	. . . 4 |- { x } = { y | x = y }
5	4	a1i 9	. . 3 |- ( x e. A -> { x } = { y | x = y } )
6	5	iuneq2i 3993	. 2 |- U_ x e. A { x } = U_ x e. A { y | x = y }
7		iunab 4022	. . 3 |- U_ x e. A { y | x = y } = { y | E. x e. A x = y }
8		risset 2561	. . . 4 |- ( y e. A <-> E. x e. A x = y )
9	8	abbii 2347	. . 3 |- { y | y e. A } = { y | E. x e. A x = y }
10		abid2 2353	. . 3 |- { y | y e. A } = A
11	7, 9, 10	3eqtr2i 2258	. 2 |- U_ x e. A { y | x = y } = A
12	6, 11	eqtri 2252	1 |- U_ x e. A { x } = A

Syntax hints:   = wceq 1398   e. wcel 2202   {cab 2217   E.wrex 2512   {csn 3673   U_ciun 3975

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-in 3207  df-ss 3214  df-sn 3679  df-iun 3977

This theorem is referenced by:  abnexg  4549  iunxpconst  4792  xpexgALT  6304  uniqs  6805