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Theorem iunid 4607
 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 4211 . . . . 5 {𝑥} = {𝑦𝑦 = 𝑥}
2 equcom 1991 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
32abbii 2768 . . . . 5 {𝑦𝑦 = 𝑥} = {𝑦𝑥 = 𝑦}
41, 3eqtri 2673 . . . 4 {𝑥} = {𝑦𝑥 = 𝑦}
54a1i 11 . . 3 (𝑥𝐴 → {𝑥} = {𝑦𝑥 = 𝑦})
65iuneq2i 4571 . 2 𝑥𝐴 {𝑥} = 𝑥𝐴 {𝑦𝑥 = 𝑦}
7 iunab 4598 . . 3 𝑥𝐴 {𝑦𝑥 = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑥 = 𝑦}
8 risset 3091 . . . 4 (𝑦𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑦)
98abbii 2768 . . 3 {𝑦𝑦𝐴} = {𝑦 ∣ ∃𝑥𝐴 𝑥 = 𝑦}
10 abid2 2774 . . 3 {𝑦𝑦𝐴} = 𝐴
117, 9, 103eqtr2i 2679 . 2 𝑥𝐴 {𝑦𝑥 = 𝑦} = 𝐴
126, 11eqtri 2673 1 𝑥𝐴 {𝑥} = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1523   ∈ wcel 2030  {cab 2637  ∃wrex 2942  {csn 4210  ∪ ciun 4552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-sn 4211  df-iun 4554 This theorem is referenced by:  iunxpconst  5209  fvn0ssdmfun  6390  abnexg  7006  xpexgALT  7203  uniqs  7850  rankcf  9637  dprd2da  18487  t1ficld  21179  discmp  21249  xkoinjcn  21538  metnrmlem2  22710  ovoliunlem1  23316  i1fima  23490  i1fd  23493  itg1addlem5  23512  sibfof  30530  bnj1415  31232  cvmlift2lem12  31422  dftrpred4g  31858  poimirlem30  33569  itg2addnclem2  33592  ftc1anclem6  33620  uniqsALTV  34242  salexct3  40878  salgensscntex  40880  ctvonmbl  41224  vonct  41228
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