Theorem abrexexg 7871

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 5237, axrep6 5236, ax-rep 5229. See also abrexex2g 7875. There are partial converses under additional conditions, see for instance abnexg 7668. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2136, ax-11 2153, ax-12 2170, ax-pr 5372, ax-un 7650 and shorten proof. (Revised by SN, 11-Dec-2024.)

Assertion

Ref	Expression
abrexexg	⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem abrexexg

Step	Hyp	Ref	Expression
1		moeq 3653	. . 3 ⊢ ∃*𝑦 𝑦 = 𝐵
2	1	ax-gen 1796	. 2 ⊢ ∀𝑥∃*𝑦 𝑦 = 𝐵
3		axrep6g 5237	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦 𝑦 = 𝐵) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
4	2, 3	mpan2 688	1 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)

Syntax hints:   → wi 4  ∀wal 1538   = wceq 1540   ∈ wcel 2105  ∃*wmo 2536  {cab 2713  ∃wrex 3070  Vcvv 3441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-rep 5229

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-v 3443

This theorem is referenced by:  abrexex  7873  iunexg  7874  qsexg  8635  wdomd  9438  cardiun  9839  rankcf  10634  sigaclci  32398  satf0suclem  33636  hbtlem1  41211  hbtlem7  41213  setpreimafvex  45186  fundcmpsurinj  45212  fundcmpsurbijinj  45213