Theorem snriota 7350

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 6480	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))})
3	1, 2	sylbi 219	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))})
4		df-rab 3394	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
5		df-riota 7317	. . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6	5	sneqi 4569	. 2 ⊢ {(℩𝑥 ∈ 𝐴 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))}
7	3, 4, 6	3eqtr4g 2801	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {(℩𝑥 ∈ 𝐴 𝜑)})

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∃!weu 2574  {cab 2719  ∃!wreu 3344  {crab 3393  {csn 4558  ℩cio 6443  ℩crio 7316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-un 3890  df-ss 3902  df-sn 4559  df-pr 4561  df-uni 4842  df-iota 6445  df-riota 7317

This theorem is referenced by:  divalgmod  16370