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Theorem riotauni 7367
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 3371 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 iotauni 6512 . . 3 (∃!𝑥(𝑥𝐴𝜑) → (℩𝑥(𝑥𝐴𝜑)) = {𝑥 ∣ (𝑥𝐴𝜑)})
31, 2sylbi 216 . 2 (∃!𝑥𝐴 𝜑 → (℩𝑥(𝑥𝐴𝜑)) = {𝑥 ∣ (𝑥𝐴𝜑)})
4 df-riota 7361 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
5 df-rab 3427 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
65unieqi 4914 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
73, 4, 63eqtr4g 2791 1 (∃!𝑥𝐴 𝜑 → (𝑥𝐴 𝜑) = {𝑥𝐴𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  ∃!weu 2556  {cab 2703  ∃!wreu 3368  {crab 3426   cuni 4902  cio 6487  crio 7360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-reu 3371  df-rab 3427  df-v 3470  df-un 3948  df-in 3950  df-ss 3960  df-sn 4624  df-pr 4626  df-uni 4903  df-iota 6489  df-riota 7361
This theorem is referenced by:  riotassuni  7402  supval2  9452  dfac2a  10126
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