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Theorem iunid 3741
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 3412 . . . . 5 {𝑥} = {𝑦𝑦 = 𝑥}
2 equcom 1634 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
32abbii 2195 . . . . 5 {𝑦𝑦 = 𝑥} = {𝑦𝑥 = 𝑦}
41, 3eqtri 2102 . . . 4 {𝑥} = {𝑦𝑥 = 𝑦}
54a1i 9 . . 3 (𝑥𝐴 → {𝑥} = {𝑦𝑥 = 𝑦})
65iuneq2i 3704 . 2 𝑥𝐴 {𝑥} = 𝑥𝐴 {𝑦𝑥 = 𝑦}
7 iunab 3732 . . 3 𝑥𝐴 {𝑦𝑥 = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑥 = 𝑦}
8 risset 2395 . . . 4 (𝑦𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑦)
98abbii 2195 . . 3 {𝑦𝑦𝐴} = {𝑦 ∣ ∃𝑥𝐴 𝑥 = 𝑦}
10 abid2 2200 . . 3 {𝑦𝑦𝐴} = 𝐴
117, 9, 103eqtr2i 2108 . 2 𝑥𝐴 {𝑦𝑥 = 𝑦} = 𝐴
126, 11eqtri 2102 1 𝑥𝐴 {𝑥} = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wcel 1434  {cab 2068  wrex 2350  {csn 3406   ciun 3686
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-in 2980  df-ss 2987  df-sn 3412  df-iun 3688
This theorem is referenced by:  iunxpconst  4426  xpexgALT  5791  uniqs  6230
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