Theorem iunid 4385

Description: An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)

Assertion

Ref	Expression
iunid

Distinct variable group:   ,

Proof of Theorem iunid

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sn 4030	. . . . 5
2		equcom 1794	. . . . . 6
3	2	abbii 2591	. . . . 5
4	1, 3	eqtri 2486	. . . 4
5	4	a1i 11	. . 3
6	5	iuneq2i 4349	. 2
7		iunab 4376	. . 3
8		risset 2982	. . . 4
9	8	abbii 2591	. . 3
10		abid2 2597	. . 3
11	7, 9, 10	3eqtr2i 2492	. 2
12	6, 11	eqtri 2486	1

Syntax hints:  =wceq 1395  e.wcel 1818  {cab 2442  E.wrex 2808  {csn 4029  U_ciun 4330

This theorem is referenced by:  iunxpconst  5061  fvn0ssdmfun  6022  xpexgALT  6793  uniqs  7390  rankcf  9176  dprd2da  17091  t1ficld  19828  discmp  19898  xkoinjcn  20188  metnrmlem2  21364  ovoliunlem1  21913  i1fima  22085  i1fd  22088  itg1addlem5  22107  sibfof  28282  cvmlift2lem12  28759  dftrpred4g  29317  itg2addnclem2  30067  ftc1anclem6  30095  bnj1415  34094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-sn 4030  df-iun 4332