Theorem riotauni 5690

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 2397	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		iotauni 5058	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
3	1, 2	sylbi 120	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
4		df-riota 5684	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5		df-rab 2399	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6	5	unieqi 3712	. 2 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ 𝜑} = ∪ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7	3, 4, 6	3eqtr4g 2172	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) = ∪ {𝑥 ∈ 𝐴 ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1314   ∈ wcel 1463  ∃!weu 1975  {cab 2101  ∃!wreu 2392  {crab 2394  ∪ cuni 3702  ℩cio 5044  ℩crio 5683

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-un 3041  df-sn 3499  df-pr 3500  df-uni 3703  df-iota 5046  df-riota 5684

This theorem is referenced by:  supval2ti  6834