Theorem iunid 3983

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 3639	. . . . 5 |- { x } = { y | y = x }
2		equcom 1729	. . . . . 6 |- ( y = x <-> x = y )
3	2	abbii 2321	. . . . 5 |- { y | y = x } = { y | x = y }
4	1, 3	eqtri 2226	. . . 4 |- { x } = { y | x = y }
5	4	a1i 9	. . 3 |- ( x e. A -> { x } = { y | x = y } )
6	5	iuneq2i 3945	. 2 |- U_ x e. A { x } = U_ x e. A { y | x = y }
7		iunab 3974	. . 3 |- U_ x e. A { y | x = y } = { y | E. x e. A x = y }
8		risset 2534	. . . 4 |- ( y e. A <-> E. x e. A x = y )
9	8	abbii 2321	. . 3 |- { y | y e. A } = { y | E. x e. A x = y }
10		abid2 2326	. . 3 |- { y | y e. A } = A
11	7, 9, 10	3eqtr2i 2232	. 2 |- U_ x e. A { y | x = y } = A
12	6, 11	eqtri 2226	1 |- U_ x e. A { x } = A

Syntax hints:   = wceq 1373   e. wcel 2176   {cab 2191   E.wrex 2485   {csn 3633   U_ciun 3927

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-sn 3639  df-iun 3929

This theorem is referenced by:  abnexg  4494  iunxpconst  4736  xpexgALT  6220  uniqs  6682