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Theorem abrexexg 7947
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 5294, axrep6 5293, ax-rep 5286. See also abrexex2g 7951. There are partial converses under additional conditions, see for instance abnexg 7743. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2138, ax-11 2155, ax-12 2172, ax-pr 5428, ax-un 7725 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 3704 . . 3 ∃*𝑦 𝑦 = 𝐵
21ax-gen 1798 . 2 𝑥∃*𝑦 𝑦 = 𝐵
3 axrep6g 5294 . 2 ((𝐴𝑉 ∧ ∀𝑥∃*𝑦 𝑦 = 𝐵) → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
42, 3mpan2 690 1 (𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540   = wceq 1542  wcel 2107  ∃*wmo 2533  {cab 2710  wrex 3071  Vcvv 3475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-rep 5286
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3072  df-v 3477
This theorem is referenced by:  abrexex  7949  iunexg  7950  qsexg  8769  wdomd  9576  cardiun  9977  rankcf  10772  sigaclci  33130  satf0suclem  34366  hbtlem1  41865  hbtlem7  41867  setpreimafvex  46051  fundcmpsurinj  46077  fundcmpsurbijinj  46078
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