Theorem abrexexg 7968

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 5295, axrep6 5294, ax-rep 5287. See also abrexex2g 7972. There are partial converses under additional conditions, see for instance abnexg 7762. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2129, ax-11 2146, ax-12 2166, ax-pr 5431, ax-un 7744 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 3702	. . 3 ⊢ ∃*𝑦 𝑦 = 𝐵
2	1	ax-gen 1789	. 2 ⊢ ∀𝑥∃*𝑦 𝑦 = 𝐵
3		axrep6g 5295	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦 𝑦 = 𝐵) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
4	2, 3	mpan2 689	1 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)

Syntax hints:   → wi 4  ∀wal 1531   = wceq 1533   ∈ wcel 2098  ∃*wmo 2527  {cab 2704  ∃wrex 3066  Vcvv 3471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-rep 5287

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-mo 2529  df-clab 2705  df-cleq 2719  df-clel 2805  df-rex 3067  df-v 3473

This theorem is referenced by:  abrexex  7970  iunexg  7971  qsexg  8798  wdomd  9610  cardiun  10011  rankcf  10806  sigaclci  33756  satf0suclem  34990  hbtlem1  42550  hbtlem7  42552  setpreimafvex  46725  fundcmpsurinj  46751  fundcmpsurbijinj  46752