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Theorem abnexg 7462
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 7648. Note that the second antecedent 𝑥𝐴𝑥𝐹 cannot be translated to 𝐴𝐹 since 𝐹 may depend on 𝑥. In applications, one may take 𝐹 = {𝑥} or 𝐹 = 𝒫 𝑥 (see snnex 7464 and pwnex 7465 respectively, proved from abnex 7463, which is a consequence of abnexg 7462 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 7450 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ 𝑊 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ V)
2 simpl 486 . . . . 5 ((𝐹𝑉𝑥𝐹) → 𝐹𝑉)
32ralimi 3131 . . . 4 (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ∀𝑥𝐴 𝐹𝑉)
4 dfiun2g 4920 . . . . . 6 (∀𝑥𝐴 𝐹𝑉 𝑥𝐴 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹})
54eleq1d 2877 . . . . 5 (∀𝑥𝐴 𝐹𝑉 → ( 𝑥𝐴 𝐹 ∈ V ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ V))
65biimprd 251 . . . 4 (∀𝑥𝐴 𝐹𝑉 → ( {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ V → 𝑥𝐴 𝐹 ∈ V))
73, 6syl 17 . . 3 (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ( {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ V → 𝑥𝐴 𝐹 ∈ V))
8 simpr 488 . . . . 5 ((𝐹𝑉𝑥𝐹) → 𝑥𝐹)
98ralimi 3131 . . . 4 (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ∀𝑥𝐴 𝑥𝐹)
10 iunid 4950 . . . . 5 𝑥𝐴 {𝑥} = 𝐴
11 snssi 4704 . . . . . . 7 (𝑥𝐹 → {𝑥} ⊆ 𝐹)
1211ralimi 3131 . . . . . 6 (∀𝑥𝐴 𝑥𝐹 → ∀𝑥𝐴 {𝑥} ⊆ 𝐹)
13 ss2iun 4902 . . . . . 6 (∀𝑥𝐴 {𝑥} ⊆ 𝐹 𝑥𝐴 {𝑥} ⊆ 𝑥𝐴 𝐹)
1412, 13syl 17 . . . . 5 (∀𝑥𝐴 𝑥𝐹 𝑥𝐴 {𝑥} ⊆ 𝑥𝐴 𝐹)
1510, 14eqsstrrid 3967 . . . 4 (∀𝑥𝐴 𝑥𝐹𝐴 𝑥𝐴 𝐹)
16 ssexg 5194 . . . . 5 ((𝐴 𝑥𝐴 𝐹 𝑥𝐴 𝐹 ∈ V) → 𝐴 ∈ V)
1716ex 416 . . . 4 (𝐴 𝑥𝐴 𝐹 → ( 𝑥𝐴 𝐹 ∈ V → 𝐴 ∈ V))
189, 15, 173syl 18 . . 3 (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ( 𝑥𝐴 𝐹 ∈ V → 𝐴 ∈ V))
197, 18syld 47 . 2 (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ( {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ V → 𝐴 ∈ V))
201, 19syl5 34 1 (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ 𝑊𝐴 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  {cab 2779  wral 3109  wrex 3110  Vcvv 3444  wss 3884  {csn 4528   cuni 4803   ciun 4884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-in 3891  df-ss 3901  df-sn 4529  df-uni 4804  df-iun 4886
This theorem is referenced by:  abnex  7463
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