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Theorem riotauni 5525
Description: Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
Assertion
Ref Expression
riotauni (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = {𝑥𝐴𝜑})

Proof of Theorem riotauni
StepHypRef Expression
1 df-reu 2360 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 iotauni 4929 . . 3 (∃!𝑥(𝑥𝐴𝜑) → (℩𝑥(𝑥𝐴𝜑)) = {𝑥 ∣ (𝑥𝐴𝜑)})
31, 2sylbi 119 . 2 (∃!𝑥𝐴 𝜑 → (℩𝑥(𝑥𝐴𝜑)) = {𝑥 ∣ (𝑥𝐴𝜑)})
4 df-riota 5519 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
5 df-rab 2362 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
65unieqi 3631 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
73, 4, 63eqtr4g 2140 1 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = {𝑥𝐴𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  ∃!weu 1943  {cab 2069  ∃!wreu 2355  {crab 2357   cuni 3621  cio 4915  crio 5518
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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-un 2986  df-sn 3422  df-pr 3423  df-uni 3622  df-iota 4917  df-riota 5519
This theorem is referenced by:  supval2ti  6502
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