Theorem iunid 4021

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 3672	. . . . 5 |- { x } = { y | y = x }
2		equcom 1752	. . . . . 6 |- ( y = x <-> x = y )
3	2	abbii 2345	. . . . 5 |- { y | y = x } = { y | x = y }
4	1, 3	eqtri 2250	. . . 4 |- { x } = { y | x = y }
5	4	a1i 9	. . 3 |- ( x e. A -> { x } = { y | x = y } )
6	5	iuneq2i 3983	. 2 |- U_ x e. A { x } = U_ x e. A { y | x = y }
7		iunab 4012	. . 3 |- U_ x e. A { y | x = y } = { y | E. x e. A x = y }
8		risset 2558	. . . 4 |- ( y e. A <-> E. x e. A x = y )
9	8	abbii 2345	. . 3 |- { y | y e. A } = { y | E. x e. A x = y }
10		abid2 2350	. . 3 |- { y | y e. A } = A
11	7, 9, 10	3eqtr2i 2256	. 2 |- U_ x e. A { y | x = y } = A
12	6, 11	eqtri 2250	1 |- U_ x e. A { x } = A

Syntax hints:   = wceq 1395   e. wcel 2200   {cab 2215   E.wrex 2509   {csn 3666   U_ciun 3965

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-sn 3672  df-iun 3967

This theorem is referenced by:  abnexg  4537  iunxpconst  4779  xpexgALT  6278  uniqs  6740