Theorem iotaexel 5975

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 5970	. . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		pm4.71r 390	. . . . . 6 ⊢ ((𝜑 → 𝑥 ∈ 𝐴) ↔ (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
3	2	albii 1518	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4		iotabi 5296	. . . . 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 2283	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥𝜑))
8		riotaexg 5974	. . 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 2309	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) → (℩𝑥𝜑) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1395   = wceq 1397   ∈ wcel 2202  Vcvv 2802  ℩cio 5284  ℩crio 5969

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-iota 5286  df-riota 5970

This theorem is referenced by: (None)