Theorem abrexexg 7986

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 5289, axrep6 5287, ax-rep 5278. See also abrexex2g 7990. There are partial converses under additional conditions, see for instance abnexg 7777. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2140, ax-11 2156, ax-12 2176, ax-pr 5431, ax-un 7756 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 3712	. . 3 ⊢ ∃*𝑦 𝑦 = 𝐵
2	1	ax-gen 1794	. 2 ⊢ ∀𝑥∃*𝑦 𝑦 = 𝐵
3		axrep6g 5289	. 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 1537   = wceq 1539   ∈ wcel 2107  ∃*wmo 2537  {cab 2713  ∃wrex 3069  Vcvv 3479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-rep 5278

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-rex 3070  df-v 3481

This theorem is referenced by:  abrexex  7988  iunexg  7989  qsexg  8816  wdomd  9622  cardiun  10023  rankcf  10818  sigaclci  34134  satf0suclem  35381  hbtlem1  43140  hbtlem7  43142  setpreimafvex  47375  fundcmpsurinj  47401  fundcmpsurbijinj  47402