Theorem iunid 3997

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 3649	. . . . 5 |- { x } = { y | y = x }
2		equcom 1730	. . . . . 6 |- ( y = x <-> x = y )
3	2	abbii 2323	. . . . 5 |- { y | y = x } = { y | x = y }
4	1, 3	eqtri 2228	. . . 4 |- { x } = { y | x = y }
5	4	a1i 9	. . 3 |- ( x e. A -> { x } = { y | x = y } )
6	5	iuneq2i 3959	. 2 |- U_ x e. A { x } = U_ x e. A { y | x = y }
7		iunab 3988	. . 3 |- U_ x e. A { y | x = y } = { y | E. x e. A x = y }
8		risset 2536	. . . 4 |- ( y e. A <-> E. x e. A x = y )
9	8	abbii 2323	. . 3 |- { y | y e. A } = { y | E. x e. A x = y }
10		abid2 2328	. . 3 |- { y | y e. A } = A
11	7, 9, 10	3eqtr2i 2234	. 2 |- U_ x e. A { y | x = y } = A
12	6, 11	eqtri 2228	1 |- U_ x e. A { x } = A

Syntax hints:   = wceq 1373   e. wcel 2178   {cab 2193   E.wrex 2487   {csn 3643   U_ciun 3941

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-sn 3649  df-iun 3943

This theorem is referenced by:  abnexg  4511  iunxpconst  4753  xpexgALT  6241  uniqs  6703