Theorem iunid 4063

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 3711	. . . . 5 |- { x } = { y | y = x }
2		equcom 1758	. . . . . 6 |- ( y = x <-> x = y )
3	2	abbii 2354	. . . . 5 |- { y | y = x } = { y | x = y }
4	1, 3	eqtri 2259	. . . 4 |- { x } = { y | x = y }
5	4	a1i 9	. . 3 |- ( x e. A -> { x } = { y | x = y } )
6	5	iuneq2i 4025	. 2 |- U_ x e. A { x } = U_ x e. A { y | x = y }
7		iunab 4054	. . 3 |- U_ x e. A { y | x = y } = { y | E. x e. A x = y }
8		risset 2578	. . . 4 |- ( y e. A <-> E. x e. A x = y )
9	8	abbii 2354	. . 3 |- { y | y e. A } = { y | E. x e. A x = y }
10		abid2 2361	. . 3 |- { y | y e. A } = A
11	7, 9, 10	3eqtr2i 2265	. 2 |- U_ x e. A { y | x = y } = A
12	6, 11	eqtri 2259	1 |- U_ x e. A { x } = A

Syntax hints:   = wceq 1402   e. wcel 2209   {cab 2224   E.wrex 2529   {csn 3705   U_ciun 4007

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-sn 3711  df-iun 4009

This theorem is referenced by:  abnexg  4587  iunxpconst  4830  xpexgALT  6356  uniqs  6857