Theorem iotaexel 5878

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 5873	. . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		pm4.71r 390	. . . . . 6 ⊢ ((𝜑 → 𝑥 ∈ 𝐴) ↔ (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
3	2	albii 1481	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4		iotabi 5224	. . . . 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 2245	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥𝜑))
8		riotaexg 5877	. . 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 2271	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) → (℩𝑥𝜑) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362   = wceq 1364   ∈ wcel 2164  Vcvv 2760  ℩cio 5213  ℩crio 5872

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-iota 5215  df-riota 5873

This theorem is referenced by: (None)