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Theorem riotauni 7230
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 3071 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 iotauni 6401 . . 3 (∃!𝑥(𝑥𝐴𝜑) → (℩𝑥(𝑥𝐴𝜑)) = {𝑥 ∣ (𝑥𝐴𝜑)})
31, 2sylbi 216 . 2 (∃!𝑥𝐴 𝜑 → (℩𝑥(𝑥𝐴𝜑)) = {𝑥 ∣ (𝑥𝐴𝜑)})
4 df-riota 7224 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
5 df-rab 3073 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
65unieqi 4852 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
73, 4, 63eqtr4g 2803 1 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = {𝑥𝐴𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  ∃!weu 2568  {cab 2715  ∃!wreu 3066  {crab 3068   cuni 4839  cio 6382  crio 7223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-reu 3071  df-rab 3073  df-v 3431  df-un 3891  df-in 3893  df-ss 3903  df-sn 4562  df-pr 4564  df-uni 4840  df-iota 6384  df-riota 7224
This theorem is referenced by:  riotassuni  7265  supval2  9201  dfac2a  9895
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