Theorem abrexexg 7910

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 5219, axrep6 5215, ax-rep 5206. See also abrexex2g 7913. There are partial converses under additional conditions, see for instance abnexg 7706. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2152, ax-11 2168, ax-12 2189, ax-pr 5369, ax-un 7685 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 3655	. . 3 ⊢ ∃*𝑦 𝑦 = 𝐵
2	1	ax-gen 1802	. 2 ⊢ ∀𝑥∃*𝑦 𝑦 = 𝐵
3		axrep6g 5219	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦 𝑦 = 𝐵) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
4	2, 3	mpan2 697	1 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)

Syntax hints:   → wi 4  ∀wal 1545   = wceq 1547   ∈ wcel 2119  ∃*wmo 2541  {cab 2718  ∃wrex 3064  Vcvv 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-rep 5206

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2719  df-cleq 2732  df-clel 2815  df-rex 3065  df-v 3434

This theorem is referenced by:  abrexex  7911  iunexg  7912  qsexg  8715  wdomd  9493  cardiun  9904  rankcf  10698  sigaclci  34323  satf0suclem  35610  hbtlem1  43575  hbtlem7  43577  setpreimafvex  47865  fundcmpsurinj  47891  fundcmpsurbijinj  47892