Theorem iunid 3928

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 3589	. . . . 5 |- { x } = { y | y = x }
2		equcom 1699	. . . . . 6 |- ( y = x <-> x = y )
3	2	abbii 2286	. . . . 5 |- { y | y = x } = { y | x = y }
4	1, 3	eqtri 2191	. . . 4 |- { x } = { y | x = y }
5	4	a1i 9	. . 3 |- ( x e. A -> { x } = { y | x = y } )
6	5	iuneq2i 3891	. 2 |- U_ x e. A { x } = U_ x e. A { y | x = y }
7		iunab 3919	. . 3 |- U_ x e. A { y | x = y } = { y | E. x e. A x = y }
8		risset 2498	. . . 4 |- ( y e. A <-> E. x e. A x = y )
9	8	abbii 2286	. . 3 |- { y | y e. A } = { y | E. x e. A x = y }
10		abid2 2291	. . 3 |- { y | y e. A } = A
11	7, 9, 10	3eqtr2i 2197	. 2 |- U_ x e. A { y | x = y } = A
12	6, 11	eqtri 2191	1 |- U_ x e. A { x } = A

Syntax hints:   = wceq 1348   e. wcel 2141   {cab 2156   E.wrex 2449   {csn 3583   U_ciun 3873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-sn 3589  df-iun 3875

This theorem is referenced by:  abnexg  4431  iunxpconst  4671  xpexgALT  6112  uniqs  6571