Theorem abnexg 7747

Description: Sufficient condition for a class abstraction to be a proper class. The class 𝐹 can be thought of as an expression in 𝑥 and the abstraction appearing in the statement as the class of values 𝐹 as 𝑥 varies through 𝐴. Assuming the antecedents, if that class is a set, then so is the "domain" 𝐴. The converse holds without antecedent, see abrexexg 7951. Note that the second antecedent ∀𝑥 ∈ 𝐴𝑥 ∈ 𝐹 cannot be translated to 𝐴 ⊆ 𝐹 since 𝐹 may depend on 𝑥. In applications, one may take 𝐹 = {𝑥} or 𝐹 = 𝒫 𝑥 (see snnex 7749 and pwnex 7750 respectively, proved from abnex 7748, which is a consequence of abnexg 7747 with 𝐴 = V). (Contributed by BJ, 2-Dec-2021.)

Assertion

Ref	Expression
abnexg	⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ 𝑊 → 𝐴 ∈ V))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐹

Allowed substitution hints:   𝐹(𝑥)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem abnexg

Step	Hyp	Ref	Expression
1		uniexg 7734	. 2 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ 𝑊 → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ V)
2		simpl 482	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → 𝐹 ∈ 𝑉)
3	2	ralimi 3082	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ∀𝑥 ∈ 𝐴 𝐹 ∈ 𝑉)
4		dfiun2g 5033	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐹 ∈ 𝑉 → ∪ 𝑥 ∈ 𝐴 𝐹 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹})
5	4	eleq1d 2817	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐹 ∈ 𝑉 → (∪ 𝑥 ∈ 𝐴 𝐹 ∈ V ↔ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ V))
6	5	biimprd 247	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐹 ∈ 𝑉 → (∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ V → ∪ 𝑥 ∈ 𝐴 𝐹 ∈ V))
7	3, 6	syl 17	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → (∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ V → ∪ 𝑥 ∈ 𝐴 𝐹 ∈ V))
8		simpr 484	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹)
9	8	ralimi 3082	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐹)
10		iunid 5063	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
11		snssi 4811	. . . . . . 7 ⊢ (𝑥 ∈ 𝐹 → {𝑥} ⊆ 𝐹)
12	11	ralimi 3082	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐹 → ∀𝑥 ∈ 𝐴 {𝑥} ⊆ 𝐹)
13		ss2iun 5015	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 {𝑥} ⊆ 𝐹 → ∪ 𝑥 ∈ 𝐴 {𝑥} ⊆ ∪ 𝑥 ∈ 𝐴 𝐹)
14	12, 13	syl 17	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐹 → ∪ 𝑥 ∈ 𝐴 {𝑥} ⊆ ∪ 𝑥 ∈ 𝐴 𝐹)
15	10, 14	eqsstrrid 4031	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐹 → 𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 𝐹)
16		ssexg 5323	. . . . 5 ⊢ ((𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 𝐹 ∧ ∪ 𝑥 ∈ 𝐴 𝐹 ∈ V) → 𝐴 ∈ V)
17	16	ex 412	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 𝐹 → (∪ 𝑥 ∈ 𝐴 𝐹 ∈ V → 𝐴 ∈ V))
18	9, 15, 17	3syl 18	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → (∪ 𝑥 ∈ 𝐴 𝐹 ∈ V → 𝐴 ∈ V))
19	7, 18	syld 47	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → (∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ V → 𝐴 ∈ V))
20	1, 19	syl5 34	1 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ 𝑊 → 𝐴 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {cab 2708  ∀wral 3060  ∃wrex 3069  Vcvv 3473   ⊆ wss 3948  {csn 4628  ∪ cuni 4908  ∪ ciun 4997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-ext 2702  ax-sep 5299  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-in 3955  df-ss 3965  df-sn 4629  df-uni 4909  df-iun 4999

This theorem is referenced by:  abnex  7748