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Theorem abrexexg 7949
Description: Existence of a class abstraction of existentially restricted sets. The class 𝐵 can be thought of as an expression in 𝑥 (which is typically a free variable in the class expression substituted for 𝐵) and the class abstraction appearing in the statement as the class of values 𝐵 as 𝑥 varies through 𝐴. If the "domain" 𝐴 is a set, then the abstraction is also a set. Therefore, this statement is a kind of Replacement. This can be seen by tracing back through the path axrep6g 5293, axrep6 5292, ax-rep 5285. See also abrexex2g 7953. There are partial converses under additional conditions, see for instance abnexg 7745. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2137, ax-11 2154, ax-12 2171, ax-pr 5427, ax-un 7727 and shorten proof. (Revised by SN, 11-Dec-2024.)
Assertion
Ref Expression
abrexexg (𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem abrexexg
StepHypRef Expression
1 moeq 3703 . . 3 ∃*𝑦 𝑦 = 𝐵
21ax-gen 1797 . 2 𝑥∃*𝑦 𝑦 = 𝐵
3 axrep6g 5293 . 2 ((𝐴𝑉 ∧ ∀𝑥∃*𝑦 𝑦 = 𝐵) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
42, 3mpan2 689 1 (𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1539   = wceq 1541  wcel 2106  ∃*wmo 2532  {cab 2709  wrex 3070  Vcvv 3474
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-rep 5285
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-rex 3071  df-v 3476
This theorem is referenced by:  abrexex  7951  iunexg  7952  qsexg  8771  wdomd  9578  cardiun  9979  rankcf  10774  sigaclci  33199  satf0suclem  34435  hbtlem1  41947  hbtlem7  41949  setpreimafvex  46130  fundcmpsurinj  46156  fundcmpsurbijinj  46157
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