Theorem iotaexab 5305

Description: Existence of the ℩ class when all the possible values are contained in a set. (Contributed by Jim Kingdon, 27-May-2025.)

Assertion

Ref	Expression
iotaexab	⊢ ({𝑥 ∣ 𝜑} ∈ 𝑉 → (℩𝑥𝜑) ∈ V)

Proof of Theorem iotaexab

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniexg 4536	. 2 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑉 → ∪ {𝑥 ∣ 𝜑} ∈ V)
2		abid 2219	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
3		elssuni 3921	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑})
4	2, 3	sylbir 135	. . . 4 ⊢ (𝜑 → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑})
5	4	ax-gen 1497	. . 3 ⊢ ∀𝑥(𝜑 → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑})
6		nfab1 2376	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
7	6	nfuni 3899	. . . . . . 7 ⊢ Ⅎ𝑥∪ {𝑥 ∣ 𝜑}
8	7	nfeq2 2386	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 = ∪ {𝑥 ∣ 𝜑}
9		sseq2 3251	. . . . . . 7 ⊢ (𝑧 = ∪ {𝑥 ∣ 𝜑} → (𝑥 ⊆ 𝑧 ↔ 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑}))
10	9	imbi2d 230	. . . . . 6 ⊢ (𝑧 = ∪ {𝑥 ∣ 𝜑} → ((𝜑 → 𝑥 ⊆ 𝑧) ↔ (𝜑 → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑})))
11	8, 10	albid 1663	. . . . 5 ⊢ (𝑧 = ∪ {𝑥 ∣ 𝜑} → (∀𝑥(𝜑 → 𝑥 ⊆ 𝑧) ↔ ∀𝑥(𝜑 → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑})))
12		sseq2 3251	. . . . 5 ⊢ (𝑧 = ∪ {𝑥 ∣ 𝜑} → ((℩𝑥𝜑) ⊆ 𝑧 ↔ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}))
13	11, 12	imbi12d 234	. . . 4 ⊢ (𝑧 = ∪ {𝑥 ∣ 𝜑} → ((∀𝑥(𝜑 → 𝑥 ⊆ 𝑧) → (℩𝑥𝜑) ⊆ 𝑧) ↔ (∀𝑥(𝜑 → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑}) → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})))
14		iotass 5304	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝑧) → (℩𝑥𝜑) ⊆ 𝑧)
15	13, 14	vtoclg 2864	. . 3 ⊢ (∪ {𝑥 ∣ 𝜑} ∈ V → (∀𝑥(𝜑 → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑}) → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}))
16	1, 5, 15	mpisyl 1491	. 2 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑉 → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
17	1, 16	ssexd 4229	1 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑉 → (℩𝑥𝜑) ∈ V)

Syntax hints:   → wi 4  ∀wal 1395   = wceq 1397   ∈ wcel 2202  {cab 2217  Vcvv 2802   ⊆ wss 3200  ∪ cuni 3893  ℩cio 5284

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

This theorem is referenced by:  fngsum  13473  igsumvalx  13474