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Theorem iunid 4548
Description: An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
Assertion
Ref Expression
iunid 𝑥𝐴 {𝑥} = 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem iunid
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-sn 4156 . . . . 5 {𝑥} = {𝑦𝑦 = 𝑥}
2 equcom 1942 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
32abbii 2736 . . . . 5 {𝑦𝑦 = 𝑥} = {𝑦𝑥 = 𝑦}
41, 3eqtri 2643 . . . 4 {𝑥} = {𝑦𝑥 = 𝑦}
54a1i 11 . . 3 (𝑥𝐴 → {𝑥} = {𝑦𝑥 = 𝑦})
65iuneq2i 4512 . 2 𝑥𝐴 {𝑥} = 𝑥𝐴 {𝑦𝑥 = 𝑦}
7 iunab 4539 . . 3 𝑥𝐴 {𝑦𝑥 = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑥 = 𝑦}
8 risset 3057 . . . 4 (𝑦𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑦)
98abbii 2736 . . 3 {𝑦𝑦𝐴} = {𝑦 ∣ ∃𝑥𝐴 𝑥 = 𝑦}
10 abid2 2742 . . 3 {𝑦𝑦𝐴} = 𝐴
117, 9, 103eqtr2i 2649 . 2 𝑥𝐴 {𝑦𝑥 = 𝑦} = 𝐴
126, 11eqtri 2643 1 𝑥𝐴 {𝑥} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  {cab 2607  wrex 2909  {csn 4155   ciun 4492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-in 3567  df-ss 3574  df-sn 4156  df-iun 4494
This theorem is referenced by:  iunxpconst  5146  fvn0ssdmfun  6316  xpexgALT  7121  uniqs  7767  rankcf  9559  dprd2da  18381  t1ficld  21071  discmp  21141  xkoinjcn  21430  metnrmlem2  22603  ovoliunlem1  23210  i1fima  23385  i1fd  23388  itg1addlem5  23407  sibfof  30225  bnj1415  30867  cvmlift2lem12  31057  dftrpred4g  31488  poimirlem30  33110  itg2addnclem2  33133  ftc1anclem6  33161  salexct3  39897  salgensscntex  39899  ctvonmbl  40240  vonct  40244
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