Theorem riotauni 5884

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 2482	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		iotauni 5231	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
3	1, 2	sylbi 121	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
4		df-riota 5877	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5		df-rab 2484	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6	5	unieqi 3849	. 2 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ 𝜑} = ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7	3, 4, 6	3eqtr4g 2254	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∪ {𝑥 ∈ 𝐴 ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃!weu 2045   ∈ wcel 2167  {cab 2182  ∃!wreu 2477  {crab 2479  ∪ cuni 3839  ℩cio 5217  ℩crio 5876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-sn 3628  df-pr 3629  df-uni 3840  df-iota 5219  df-riota 5877

This theorem is referenced by:  supval2ti  7061