Theorem iotaexel 6033

Description: Set existence of an iota expression in which all values are contained within a set. (Contributed by Jim Kingdon, 28-Jun-2025.)

Assertion

Ref	Expression
iotaexel	⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) → (℩𝑥𝜑) ∈ V)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem iotaexel

Step	Hyp	Ref	Expression
1		df-riota 6028	. . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		pm4.71r 394	. . . . . 6 ⊢ ((𝜑 → 𝑥 ∈ 𝐴) ↔ (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
3	2	albii 1523	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4		iotabi 5342	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) → (℩𝑥𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
5	3, 4	sylbi 121	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) → (℩𝑥𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
6	5	adantl 277	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) → (℩𝑥𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
7	1, 6	eqtr4id 2290	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥𝜑))
8		riotaexg 6032	. . 3 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥 ∈ 𝐴 𝜑) ∈ V)
9	8	adantr 276	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) → (℩𝑥 ∈ 𝐴 𝜑) ∈ V)
10	7, 9	eqeltrrd 2316	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) → (℩𝑥𝜑) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1400   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ℩cio 5330  ℩crio 6027

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-14 2212  ax-ext 2220  ax-sep 4244  ax-un 4573

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-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-iota 5332  df-riota 6028

This theorem is referenced by: (None)