Theorem abnexg 4567

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 6311. 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 4569 and pwnex 4570 respectively, proved from abnex 4568, which is a consequence of abnexg 4567 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 4560	. 2 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ 𝑊 → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ V)
2		simpl 109	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → 𝐹 ∈ 𝑉)
3	2	ralimi 2605	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ∀𝑥 ∈ 𝐴 𝐹 ∈ 𝑉)
4		dfiun2g 4023	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐹 ∈ 𝑉 → ∪ 𝑥 ∈ 𝐴 𝐹 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹})
5	4	eleq1d 2301	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐹 ∈ 𝑉 → (∪ 𝑥 ∈ 𝐴 𝐹 ∈ V ↔ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ V))
6	5	biimprd 158	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐹 ∈ 𝑉 → (∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ V → ∪ 𝑥 ∈ 𝐴 𝐹 ∈ V))
7	3, 6	syl 14	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → (∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ V → ∪ 𝑥 ∈ 𝐴 𝐹 ∈ V))
8		simpr 110	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹)
9	8	ralimi 2605	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐹)
10		iunid 4047	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
11		snssi 3838	. . . . . . 7 ⊢ (𝑥 ∈ 𝐹 → {𝑥} ⊆ 𝐹)
12	11	ralimi 2605	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐹 → ∀𝑥 ∈ 𝐴 {𝑥} ⊆ 𝐹)
13		ss2iun 4006	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 {𝑥} ⊆ 𝐹 → ∪ 𝑥 ∈ 𝐴 {𝑥} ⊆ ∪ 𝑥 ∈ 𝐴 𝐹)
14	12, 13	syl 14	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐹 → ∪ 𝑥 ∈ 𝐴 {𝑥} ⊆ ∪ 𝑥 ∈ 𝐴 𝐹)
15	10, 14	eqsstrrid 3285	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐹 → 𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 𝐹)
16		ssexg 4249	. . . . 5 ⊢ ((𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 𝐹 ∧ ∪ 𝑥 ∈ 𝐴 𝐹 ∈ V) → 𝐴 ∈ V)
17	16	ex 115	. . . 4 ⊢ (𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 𝐹 → (∪ 𝑥 ∈ 𝐴 𝐹 ∈ V → 𝐴 ∈ V))
18	9, 15, 17	3syl 17	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → (∪ 𝑥 ∈ 𝐴 𝐹 ∈ V → 𝐴 ∈ V))
19	7, 18	syld 45	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → (∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ V → 𝐴 ∈ V))
20	1, 19	syl5 32	1 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ 𝑊 → 𝐴 ∈ V))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  {cab 2218  ∀wral 2520  ∃wrex 2521  Vcvv 2813   ⊆ wss 3211  {csn 3689  ∪ cuni 3914  ∪ ciun 3991

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-in 3217  df-ss 3224  df-sn 3695  df-uni 3915  df-iun 3993

This theorem is referenced by:  abnex  4568