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Theorem uniiun 3834
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 3706 . 2 𝐴 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝑥}
2 df-iun 3783 . 2 𝑥𝐴 𝑥 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝑥}
31, 2eqtr4i 2139 1 𝐴 = 𝑥𝐴 𝑥
Colors of variables: wff set class
Syntax hints:   = wceq 1314  {cab 2101  wrex 2392   cuni 3704   ciun 3781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-rex 2397  df-uni 3705  df-iun 3783
This theorem is referenced by:  iunpwss  3872  truni  4008  iunpw  4369  reluni  4630  rnuni  4918  imauni  5628  hashuni  11202  tgidm  12149  unicld  12191  tgrest  12244  txbasval  12342
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