Theorem reuabaiotaiota 44579

Description: The iota and the alternate iota over a wff 𝜑 are equal iff there is a unique satisfying value of {𝑥 ∣ 𝜑} = {𝑦}. (Contributed by AV, 25-Aug-2022.)

Assertion

Ref	Expression
reuabaiotaiota	⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reuabaiotaiota

Step	Hyp	Ref	Expression
1		uniintab 4919	. 2 ⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}})
2		df-iota 6391	. . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
3		df-aiota 44577	. . 3 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
4	2, 3	eqeq12i 2756	. 2 ⊢ ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}})
5	1, 4	bitr4i 277	1 ⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))

Syntax hints:   ↔ wb 205   = wceq 1539  ∃!weu 2568  {cab 2715  {csn 4561  ∪ cuni 4839  ∩ cint 4879  ℩cio 6389  ℩'caiota 44575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-pr 4564  df-uni 4840  df-int 4880  df-iota 6391  df-aiota 44577

This theorem is referenced by:  reuaiotaiota  44580  aiotaval  44587