Theorem iotaexab 5307

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 4538	. 2 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑉 → ∪ {𝑥 ∣ 𝜑} ∈ V)
2		abid 2218	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
3		elssuni 3922	. . . . 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 2375	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
7	6	nfuni 3900	. . . . . . 7 ⊢ Ⅎ𝑥∪ {𝑥 ∣ 𝜑}
8	7	nfeq2 2385	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 = ∪ {𝑥 ∣ 𝜑}
9		sseq2 3250	. . . . . . 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 3250	. . . . 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 5306	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝑧) → (℩𝑥𝜑) ⊆ 𝑧)
15	13, 14	vtoclg 2863	. . 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 4230	1 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑉 → (℩𝑥𝜑) ∈ V)

Syntax hints:   → wi 4  ∀wal 1395   = wceq 1397   ∈ wcel 2201  {cab 2216  Vcvv 2801   ⊆ wss 3199  ∪ cuni 3894  ℩cio 5286

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 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-un 4532

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-uni 3895  df-iota 5288

This theorem is referenced by:  fngsum  13494  igsumvalx  13495