Theorem iotaexab 5273

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 4507	. 2 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑉 → ∪ {𝑥 ∣ 𝜑} ∈ V)
2		abid 2197	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
3		elssuni 3895	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑})
4	2, 3	sylbir 135	. . . 4 ⊢ (𝜑 → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑})
5	4	ax-gen 1475	. . 3 ⊢ ∀𝑥(𝜑 → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑})
6		nfab1 2354	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
7	6	nfuni 3873	. . . . . . 7 ⊢ Ⅎ𝑥∪ {𝑥 ∣ 𝜑}
8	7	nfeq2 2364	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 = ∪ {𝑥 ∣ 𝜑}
9		sseq2 3228	. . . . . . 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 1641	. . . . 5 ⊢ (𝑧 = ∪ {𝑥 ∣ 𝜑} → (∀𝑥(𝜑 → 𝑥 ⊆ 𝑧) ↔ ∀𝑥(𝜑 → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑})))
12		sseq2 3228	. . . . 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 5272	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝑧) → (℩𝑥𝜑) ⊆ 𝑧)
15	13, 14	vtoclg 2841	. . 3 ⊢ (∪ {𝑥 ∣ 𝜑} ∈ V → (∀𝑥(𝜑 → 𝑥 ⊆ ∪ {𝑥 ∣ 𝜑}) → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}))
16	1, 5, 15	mpisyl 1469	. 2 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑉 → (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑})
17	1, 16	ssexd 4203	1 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝑉 → (℩𝑥𝜑) ∈ V)

Syntax hints:   → wi 4  ∀wal 1373   = wceq 1375   ∈ wcel 2180  {cab 2195  Vcvv 2779   ⊆ wss 3177  ∪ cuni 3867  ℩cio 5252

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-un 4501

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-uni 3868  df-iota 5254

This theorem is referenced by:  fngsum  13387  igsumvalx  13388