Theorem abnexg 4431

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 6097. 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 4433 and pwnex 4434 respectively, proved from abnex 4432, which is a consequence of abnexg 4431 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 4424	. 2 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ 𝑊 → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ V)
2		simpl 108	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → 𝐹 ∈ 𝑉)
3	2	ralimi 2533	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ∀𝑥 ∈ 𝐴 𝐹 ∈ 𝑉)
4		dfiun2g 3905	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐹 ∈ 𝑉 → ∪ 𝑥 ∈ 𝐴 𝐹 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹})
5	4	eleq1d 2239	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐹 ∈ 𝑉 → (∪ 𝑥 ∈ 𝐴 𝐹 ∈ V ↔ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ V))
6	5	biimprd 157	. . . 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 109	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹)
9	8	ralimi 2533	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐹)
10		iunid 3928	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
11		snssi 3724	. . . . . . 7 ⊢ (𝑥 ∈ 𝐹 → {𝑥} ⊆ 𝐹)
12	11	ralimi 2533	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐹 → ∀𝑥 ∈ 𝐴 {𝑥} ⊆ 𝐹)
13		ss2iun 3888	. . . . . 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 3194	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐹 → 𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 𝐹)
16		ssexg 4128	. . . . 5 ⊢ ((𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 𝐹 ∧ ∪ 𝑥 ∈ 𝐴 𝐹 ∈ V) → 𝐴 ∈ V)
17	16	ex 114	. . . 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 103   = wceq 1348   ∈ wcel 2141  {cab 2156  ∀wral 2448  ∃wrex 2449  Vcvv 2730   ⊆ wss 3121  {csn 3583  ∪ cuni 3796  ∪ ciun 3873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-sn 3589  df-uni 3797  df-iun 3875

This theorem is referenced by:  abnex  4432