Theorem reuabaiotaiota 47532

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 4929	. 2 ⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}})
2		df-iota 6446	. . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
3		df-aiota 47530	. . 3 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
4	2, 3	eqeq12i 2755	. 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 1542  ∃!weu 2569  {cab 2715  {csn 4568  ∪ cuni 4851  ∩ cint 4890  ℩cio 6444  ℩'caiota 47528

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 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-sn 4569  df-pr 4571  df-uni 4852  df-int 4891  df-iota 6446  df-aiota 47530

This theorem is referenced by:  reuaiotaiota  47533  aiotaval  47540