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Theorem abnexg 7129
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 7305. 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 7131 and pwnex 7133 respectively, proved from abnex 7130, which is a consequence of abnexg 7129 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 7120 . 2 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ 𝑊 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ V)
2 simpl 474 . . . . 5 ((𝐹𝑉𝑥𝐹) → 𝐹𝑉)
32ralimi 3090 . . . 4 (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ∀𝑥𝐴 𝐹𝑉)
4 dfiun2g 4704 . . . . . 6 (∀𝑥𝐴 𝐹𝑉 𝑥𝐴 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹})
54eleq1d 2824 . . . . 5 (∀𝑥𝐴 𝐹𝑉 → ( 𝑥𝐴 𝐹 ∈ V ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ V))
65biimprd 238 . . . 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 479 . . . . 5 ((𝐹𝑉𝑥𝐹) → 𝑥𝐹)
98ralimi 3090 . . . 4 (∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ∀𝑥𝐴 𝑥𝐹)
10 iunid 4727 . . . . 5 𝑥𝐴 {𝑥} = 𝐴
11 snssi 4484 . . . . . . 7 (𝑥𝐹 → {𝑥} ⊆ 𝐹)
1211ralimi 3090 . . . . . 6 (∀𝑥𝐴 𝑥𝐹 → ∀𝑥𝐴 {𝑥} ⊆ 𝐹)
13 ss2iun 4688 . . . . . 6 (∀𝑥𝐴 {𝑥} ⊆ 𝐹 𝑥𝐴 {𝑥} ⊆ 𝑥𝐴 𝐹)
1412, 13syl 17 . . . . 5 (∀𝑥𝐴 𝑥𝐹 𝑥𝐴 {𝑥} ⊆ 𝑥𝐴 𝐹)
1510, 14syl5eqssr 3791 . . . 4 (∀𝑥𝐴 𝑥𝐹𝐴 𝑥𝐴 𝐹)
16 ssexg 4956 . . . . 5 ((𝐴 𝑥𝐴 𝐹 𝑥𝐴 𝐹 ∈ V) → 𝐴 ∈ V)
1716ex 449 . . . 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 383   = wceq 1632  wcel 2139  {cab 2746  wral 3050  wrex 3051  Vcvv 3340  wss 3715  {csn 4321   cuni 4588   ciun 4672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-sn 4322  df-uni 4589  df-iun 4674
This theorem is referenced by:  abnex  7130
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