Theorem eliota 5360

Description: An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.)

Assertion

Ref	Expression
eliota	⊢ (𝐴 ∈ (℩𝑥𝜑) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))

Distinct variable groups:   𝜑,𝑦   𝑦,𝐴   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem eliota

Step	Hyp	Ref	Expression
1		dfiota2 5333	. . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
2	1	eleq2i 2305	. 2 ⊢ (𝐴 ∈ (℩𝑥𝜑) ↔ 𝐴 ∈ ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)})
3		eluniab 3942	. 2 ⊢ (𝐴 ∈ ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)} ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
4	2, 3	bitri 184	1 ⊢ (𝐴 ∈ (℩𝑥𝜑) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1400  ∃wex 1545   ∈ wcel 2209  {cab 2224  ∪ cuni 3930  ℩cio 5330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sn 3711  df-uni 3931  df-iota 5332

This theorem is referenced by:  eliotaeu  5361