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Theorem iunid 3780
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 3447 . . . . 5 {𝑥} = {𝑦𝑦 = 𝑥}
2 equcom 1639 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
32abbii 2203 . . . . 5 {𝑦𝑦 = 𝑥} = {𝑦𝑥 = 𝑦}
41, 3eqtri 2108 . . . 4 {𝑥} = {𝑦𝑥 = 𝑦}
54a1i 9 . . 3 (𝑥𝐴 → {𝑥} = {𝑦𝑥 = 𝑦})
65iuneq2i 3743 . 2 𝑥𝐴 {𝑥} = 𝑥𝐴 {𝑦𝑥 = 𝑦}
7 iunab 3771 . . 3 𝑥𝐴 {𝑦𝑥 = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑥 = 𝑦}
8 risset 2406 . . . 4 (𝑦𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑦)
98abbii 2203 . . 3 {𝑦𝑦𝐴} = {𝑦 ∣ ∃𝑥𝐴 𝑥 = 𝑦}
10 abid2 2208 . . 3 {𝑦𝑦𝐴} = 𝐴
117, 9, 103eqtr2i 2114 . 2 𝑥𝐴 {𝑦𝑥 = 𝑦} = 𝐴
126, 11eqtri 2108 1 𝑥𝐴 {𝑥} = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wcel 1438  {cab 2074  wrex 2360  {csn 3441   ciun 3725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3003  df-ss 3010  df-sn 3447  df-iun 3727
This theorem is referenced by:  iunxpconst  4486  xpexgALT  5886  uniqs  6330
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