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Theorem riotauni 5417
Description: Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
Assertion
Ref Expression
riotauni (∃!x A φ → (x A φ) = {x Aφ})

Proof of Theorem riotauni
StepHypRef Expression
1 df-reu 2307 . . 3 (∃!x A φ∃!x(x A φ))
2 iotauni 4822 . . 3 (∃!x(x A φ) → (℩x(x A φ)) = {x ∣ (x A φ)})
31, 2sylbi 114 . 2 (∃!x A φ → (℩x(x A φ)) = {x ∣ (x A φ)})
4 df-riota 5411 . 2 (x A φ) = (℩x(x A φ))
5 df-rab 2309 . . 3 {x Aφ} = {x ∣ (x A φ)}
65unieqi 3581 . 2 {x Aφ} = {x ∣ (x A φ)}
73, 4, 63eqtr4g 2094 1 (∃!x A φ → (x A φ) = {x Aφ})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  ∃!weu 1897  {cab 2023  ∃!wreu 2302  {crab 2304   cuni 3571  cio 4808  crio 5410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810  df-riota 5411
This theorem is referenced by: (None)
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