Theorem iotaexel 6016

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 6011	. . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		pm4.71r 390	. . . . . 6 ⊢ ((𝜑 → 𝑥 ∈ 𝐴) ↔ (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
3	2	albii 1519	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4		iotabi 5327	. . . . 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 2286	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥𝜑))
8		riotaexg 6015	. . 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 2312	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) → (℩𝑥𝜑) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1396   = wceq 1398   ∈ wcel 2205  Vcvv 2815  ℩cio 5315  ℩crio 6010

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-iota 5317  df-riota 6011

This theorem is referenced by: (None)