Theorem snriota 7336

Description: A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)

Assertion

Ref	Expression
snriota	⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {(℩𝑥 ∈ 𝐴 𝜑)})

Proof of Theorem snriota

Step	Hyp	Ref	Expression
1		df-reu 3347	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		sniota 6472	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))})
3	1, 2	sylbi 217	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))})
4		df-rab 3396	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
5		df-riota 7303	. . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6	5	sneqi 4584	. 2 ⊢ {(℩𝑥 ∈ 𝐴 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))}
7	3, 4, 6	3eqtr4g 2791	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {(℩𝑥 ∈ 𝐴 𝜑)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃!weu 2563  {cab 2709  ∃!wreu 3344  {crab 3395  {csn 4573  ℩cio 6435  ℩crio 7302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-un 3902  df-ss 3914  df-sn 4574  df-pr 4576  df-uni 4857  df-iota 6437  df-riota 7303

This theorem is referenced by:  divalgmod  16317