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Theorem iunid 4975
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 4558 . . . . 5 {𝑥} = {𝑦𝑦 = 𝑥}
2 equcom 2016 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
32abbii 2883 . . . . 5 {𝑦𝑦 = 𝑥} = {𝑦𝑥 = 𝑦}
41, 3eqtri 2841 . . . 4 {𝑥} = {𝑦𝑥 = 𝑦}
54a1i 11 . . 3 (𝑥𝐴 → {𝑥} = {𝑦𝑥 = 𝑦})
65iuneq2i 4931 . 2 𝑥𝐴 {𝑥} = 𝑥𝐴 {𝑦𝑥 = 𝑦}
7 iunab 4966 . . 3 𝑥𝐴 {𝑦𝑥 = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑥 = 𝑦}
8 risset 3264 . . . 4 (𝑦𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑦)
98abbii 2883 . . 3 {𝑦𝑦𝐴} = {𝑦 ∣ ∃𝑥𝐴 𝑥 = 𝑦}
10 abid2 2954 . . 3 {𝑦𝑦𝐴} = 𝐴
117, 9, 103eqtr2i 2847 . 2 𝑥𝐴 {𝑦𝑥 = 𝑦} = 𝐴
126, 11eqtri 2841 1 𝑥𝐴 {𝑥} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  {cab 2796  wrex 3136  {csn 4557   ciun 4910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-in 3940  df-ss 3949  df-sn 4558  df-iun 4912
This theorem is referenced by:  iunxpconst  5617  fvn0ssdmfun  6834  abnexg  7467  xpexgALT  7671  uniqs  8346  rankcf  10187  dprd2da  19093  t1ficld  21863  discmp  21934  xkoinjcn  22223  metnrmlem2  23395  ovoliunlem1  24030  i1fima  24206  i1fd  24209  itg1addlem5  24228  fnpreimac  30344  sibfof  31497  bnj1415  32207  cvmlift2lem12  32458  dftrpred4g  32970  poimirlem30  34803  itg2addnclem2  34825  ftc1anclem6  34853  uniqsALTV  35467  salexct3  42502  salgensscntex  42504  ctvonmbl  42848  vonct  42852
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