Theorem iunid 3712

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 3381	. . . . 5 |- { x } = { y | y = x }
2		equcom 1593	. . . . . 6 |- ( y = x <-> x = y )
3	2	abbii 2153	. . . . 5 |- { y | y = x } = { y | x = y }
4	1, 3	eqtri 2060	. . . 4 |- { x } = { y | x = y }
5	4	a1i 9	. . 3 |- ( x e. A -> { x } = { y | x = y } )
6	5	iuneq2i 3675	. 2 |- U_ x e. A { x } = U_ x e. A { y | x = y }
7		iunab 3703	. . 3 |- U_ x e. A { y | x = y } = { y | E. x e. A x = y }
8		risset 2352	. . . 4 |- ( y e. A <-> E. x e. A x = y )
9	8	abbii 2153	. . 3 |- { y | y e. A } = { y | E. x e. A x = y }
10		abid2 2158	. . 3 |- { y | y e. A } = A
11	7, 9, 10	3eqtr2i 2066	. 2 |- U_ x e. A { y | x = y } = A
12	6, 11	eqtri 2060	1 |- U_ x e. A { x } = A

Syntax hints:   = wceq 1243   e. wcel 1393   {cab 2026   E.wrex 2307   {csn 3375   U_ciun 3657

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-sn 3381  df-iun 3659

This theorem is referenced by:  iunxpconst  4400  xpexgALT  5760  uniqs  6164