Theorem reuabaiotaiota 47092

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 4953	. 2 ⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}})
2		df-iota 6467	. . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
3		df-aiota 47090	. . 3 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
4	2, 3	eqeq12i 2748	. 2 ⊢ ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}})
5	1, 4	bitr4i 278	1 ⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))

Syntax hints:   ↔ wb 206   = wceq 1540  ∃!weu 2562  {cab 2708  {csn 4592  ∪ cuni 4874  ∩ cint 4913  ℩cio 6465  ℩'caiota 47088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-sn 4593  df-pr 4595  df-uni 4875  df-int 4914  df-iota 6467  df-aiota 47090

This theorem is referenced by:  reuaiotaiota  47093  aiotaval  47100