Theorem riotauni 5494

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 ∈ 𝐴 ∣ 𝜑})

Proof of Theorem riotauni

Step	Hyp	Ref	Expression
1		df-reu 2355	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		iotauni 4899	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
3	1, 2	sylbi 119	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
4		df-riota 5488	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5		df-rab 2357	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6	5	unieqi 3611	. 2 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ 𝜑} = ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7	3, 4, 6	3eqtr4g 2138	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∪ {𝑥 ∈ 𝐴 ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  ∃!weu 1941  {cab 2067  ∃!wreu 2350  {crab 2352  ∪ cuni 3601  ℩cio 4885  ℩crio 5487

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602  df-iota 4887  df-riota 5488

This theorem is referenced by:  supval2ti  6408