Theorem iotaexab 5214

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 4457	. 2 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑉 → ∪ {𝑥 ∣ 𝜑} ∈ V)
2		abid 2177	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
3		elssuni 3852	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑})
4	2, 3	sylbir 135	. . . 4 ⊢ (𝜑 → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑})
5	4	ax-gen 1460	. . 3 ⊢ ∀𝑥(𝜑 → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑})
6		nfab1 2334	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
7	6	nfuni 3830	. . . . . . 7 ⊢ Ⅎ𝑥∪ {𝑥 ∣ 𝜑}
8	7	nfeq2 2344	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 = ∪ {𝑥 ∣ 𝜑}
9		sseq2 3194	. . . . . . 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 1626	. . . . 5 ⊢ (𝑧 = ∪ {𝑥 ∣ 𝜑} → (∀𝑥(𝜑 → 𝑥 ⊆ 𝑧) ↔ ∀𝑥(𝜑 → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑})))
12		sseq2 3194	. . . . 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 5213	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝑧) → (℩𝑥𝜑) ⊆ 𝑧)
15	13, 14	vtoclg 2812	. . 3 ⊢ (∪ {𝑥 ∣ 𝜑} ∈ V → (∀𝑥(𝜑 → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑}) → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}))
16	1, 5, 15	mpisyl 1457	. 2 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑉 → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
17	1, 16	ssexd 4158	1 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑉 → (℩𝑥𝜑) ∈ V)

Syntax hints:   → wi 4  ∀wal 1362   = wceq 1364   ∈ wcel 2160  {cab 2175  Vcvv 2752   ⊆ wss 3144  ∪ cuni 3824  ℩cio 5194

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-iota 5196

This theorem is referenced by:  fngsum  12867  igsumvalx  12868