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Theorem uniiun 3778
Description: Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)
Assertion
Ref Expression
uniiun 𝐴 = 𝑥𝐴 𝑥
Distinct variable group:   𝑥,𝐴

Proof of Theorem uniiun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfuni2 3650 . 2 𝐴 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝑥}
2 df-iun 3727 . 2 𝑥𝐴 𝑥 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝑥}
31, 2eqtr4i 2111 1 𝐴 = 𝑥𝐴 𝑥
Colors of variables: wff set class
Syntax hints:   = wceq 1289  {cab 2074  wrex 2360   cuni 3648   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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  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-rex 2365  df-uni 3649  df-iun 3727
This theorem is referenced by:  iunpwss  3812  truni  3942  iunpw  4292  reluni  4548  rnuni  4830  imauni  5522  hashuni  10838
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