Theorem riotauni 7388

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 3375	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		iotauni 6528	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
3	1, 2	sylbi 216	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
4		df-riota 7382	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5		df-rab 3431	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6	5	unieqi 4924	. 2 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ 𝜑} = ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7	3, 4, 6	3eqtr4g 2793	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∪ {𝑥 ∈ 𝐴 ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃!weu 2557  {cab 2705  ∃!wreu 3372  {crab 3430  ∪ cuni 4912  ℩cio 6503  ℩crio 7381

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 2166  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-reu 3375  df-rab 3431  df-v 3475  df-un 3954  df-in 3956  df-ss 3966  df-sn 4633  df-pr 4635  df-uni 4913  df-iota 6505  df-riota 7382

This theorem is referenced by:  riotassuni  7423  supval2  9486  dfac2a  10160