Theorem aiotaexb 47039

Description: The alternate iota over a wff 𝜑 is a set iff there is a unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.)

Assertion

Ref	Expression
aiotaexb	⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)

Proof of Theorem aiotaexb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		intexab 5352	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
2		euabsn2 4730	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
3		df-aiota 47035	. . 3 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
4	3	eleq1i 2830	. 2 ⊢ ((℩'𝑥𝜑) ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
5	1, 2, 4	3bitr4i 303	1 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)

Syntax hints:   ↔ wb 206   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ∃!weu 2566  {cab 2712  Vcvv 3478  {csn 4631  ∩ cint 4951  ℩'caiota 47033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-int 4952  df-aiota 47035

This theorem is referenced by:  aiotavb  47040  iotan0aiotaex  47043  aiotaexaiotaiota  47044  aiota0ndef  47047