Theorem abrexexg 7938

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 5239, axrep6 5235, ax-rep 5226. See also abrexex2g 7941. There are partial converses under additional conditions, see for instance abnexg 7735. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2174, ax-11 2190, ax-12 2211, ax-pr 5389, ax-un 7714 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 3669	. . 3 ⊢ ∃*𝑦 𝑦 = 𝐵
2	1	ax-gen 1814	. 2 ⊢ ∀𝑥∃*𝑦 𝑦 = 𝐵
3		axrep6g 5239	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦 𝑦 = 𝐵) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
4	2, 3	mpan2 701	1 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)

Syntax hints:   → wi 4  ∀wal 1557   = wceq 1559   ∈ wcel 2141  ∃*wmo 2563  {cab 2739  ∃wrex 3085  Vcvv 3453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-rep 5226

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-rex 3086  df-v 3455

This theorem is referenced by:  abrexex  7939  iunexg  7940  qsexg  8748  wdomd  9526  cardiun  9937  rankcf  10732  sigaclci  34390  satf0suclem  35689  hbtlem1  43664  hbtlem7  43666  setpreimafvex  47953  fundcmpsurinj  47979  fundcmpsurbijinj  47980