Theorem aiotaexb 47537

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 5287	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
2		euabsn2 4669	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
3		df-aiota 47533	. . 3 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
4	3	eleq1i 2827	. 2 ⊢ ((℩'𝑥𝜑) ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
5	1, 2, 4	3bitr4i 303	1 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)

Syntax hints:   ↔ wb 206   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃!weu 2568  {cab 2714  Vcvv 3429  {csn 4567  ∩ cint 4889  ℩'caiota 47531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-in 3896  df-ss 3906  df-nul 4274  df-sn 4568  df-int 4890  df-aiota 47533

This theorem is referenced by:  aiotavb  47538  iotan0aiotaex  47541  aiotaexaiotaiota  47542  aiota0ndef  47545