Theorem aiotaexb 47680

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 5302	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
2		euabsn2 4684	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
3		df-aiota 47676	. . 3 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
4	3	eleq1i 2853	. 2 ⊢ ((℩'𝑥𝜑) ∈ V ↔ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ∈ V)
5	1, 2, 4	3bitr4i 305	1 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)

Syntax hints:   ↔ wb 208   = wceq 1560  ∃wex 1799   ∈ wcel 2142  ∃!weu 2595  {cab 2740  Vcvv 3454  {csn 4582  ∩ cint 4905  ℩'caiota 47674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-in 3911  df-ss 3921  df-nul 4286  df-sn 4583  df-int 4906  df-aiota 47676

This theorem is referenced by:  aiotavb  47681  iotan0aiotaex  47684  aiotaexaiotaiota  47685  aiota0ndef  47688