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Theorem abnexg 4305
 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 5947. 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 4307 and pwnex 4308 respectively, proved from abnex 4306, which is a consequence of abnexg 4305 with 𝐴 = V). (Contributed by BJ, 2-Dec-2021.)
Assertion
Ref Expression
abnexg (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ 𝑊𝐴 ∈ V))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐹
Allowed substitution hints:   𝐹(𝑥)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem abnexg
StepHypRef Expression
1 uniexg 4299 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ 𝑊 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ V)
2 simpl 108 . . . . 5 ((𝐹𝑉𝑥𝐹) → 𝐹𝑉)
32ralimi 2454 . . . 4 (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ∀𝑥𝐴 𝐹𝑉)
4 dfiun2g 3792 . . . . . 6 (∀𝑥𝐴 𝐹𝑉 𝑥𝐴 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹})
54eleq1d 2168 . . . . 5 (∀𝑥𝐴 𝐹𝑉 → ( 𝑥𝐴 𝐹 ∈ V ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ V))
65biimprd 157 . . . 4 (∀𝑥𝐴 𝐹𝑉 → ( {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ V → 𝑥𝐴 𝐹 ∈ V))
73, 6syl 14 . . 3 (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ( {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ V → 𝑥𝐴 𝐹 ∈ V))
8 simpr 109 . . . . 5 ((𝐹𝑉𝑥𝐹) → 𝑥𝐹)
98ralimi 2454 . . . 4 (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ∀𝑥𝐴 𝑥𝐹)
10 iunid 3815 . . . . 5 𝑥𝐴 {𝑥} = 𝐴
11 snssi 3611 . . . . . . 7 (𝑥𝐹 → {𝑥} ⊆ 𝐹)
1211ralimi 2454 . . . . . 6 (∀𝑥𝐴 𝑥𝐹 → ∀𝑥𝐴 {𝑥} ⊆ 𝐹)
13 ss2iun 3775 . . . . . 6 (∀𝑥𝐴 {𝑥} ⊆ 𝐹 𝑥𝐴 {𝑥} ⊆ 𝑥𝐴 𝐹)
1412, 13syl 14 . . . . 5 (∀𝑥𝐴 𝑥𝐹 𝑥𝐴 {𝑥} ⊆ 𝑥𝐴 𝐹)
1510, 14syl5eqssr 3094 . . . 4 (∀𝑥𝐴 𝑥𝐹𝐴 𝑥𝐴 𝐹)
16 ssexg 4007 . . . . 5 ((𝐴 𝑥𝐴 𝐹 𝑥𝐴 𝐹 ∈ V) → 𝐴 ∈ V)
1716ex 114 . . . 4 (𝐴 𝑥𝐴 𝐹 → ( 𝑥𝐴 𝐹 ∈ V → 𝐴 ∈ V))
189, 15, 173syl 17 . . 3 (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ( 𝑥𝐴 𝐹 ∈ V → 𝐴 ∈ V))
197, 18syld 45 . 2 (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ( {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ V → 𝐴 ∈ V))
201, 19syl5 32 1 (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ 𝑊𝐴 ∈ V))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1299   ∈ wcel 1448  {cab 2086  ∀wral 2375  ∃wrex 2376  Vcvv 2641   ⊆ wss 3021  {csn 3474  ∪ cuni 3683  ∪ ciun 3760 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-un 4293 This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-in 3027  df-ss 3034  df-sn 3480  df-uni 3684  df-iun 3762 This theorem is referenced by:  abnex  4306
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