Theorem abrexexg 7964

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 5265, axrep6 5263, ax-rep 5254. See also abrexex2g 7968. There are partial converses under additional conditions, see for instance abnexg 7755. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2142, ax-11 2158, ax-12 2178, ax-pr 5407, ax-un 7734 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 3695	. . 3 ⊢ ∃*𝑦 𝑦 = 𝐵
2	1	ax-gen 1795	. 2 ⊢ ∀𝑥∃*𝑦 𝑦 = 𝐵
3		axrep6g 5265	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦 𝑦 = 𝐵) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
4	2, 3	mpan2 691	1 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)

Syntax hints:   → wi 4  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∃*wmo 2538  {cab 2714  ∃wrex 3061  Vcvv 3464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-rep 5254

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-mo 2540  df-clab 2715  df-cleq 2728  df-clel 2810  df-rex 3062  df-v 3466

This theorem is referenced by:  abrexex  7966  iunexg  7967  qsexg  8794  wdomd  9600  cardiun  10001  rankcf  10796  sigaclci  34168  satf0suclem  35402  hbtlem1  43114  hbtlem7  43116  setpreimafvex  47364  fundcmpsurinj  47390  fundcmpsurbijinj  47391