Theorem abrexexg 7984

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 5296, axrep6 5294, ax-rep 5285. See also abrexex2g 7988. There are partial converses under additional conditions, see for instance abnexg 7775. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2139, ax-11 2155, ax-12 2175, ax-pr 5438, ax-un 7754 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 3716	. . 3 ⊢ ∃*𝑦 𝑦 = 𝐵
2	1	ax-gen 1792	. 2 ⊢ ∀𝑥∃*𝑦 𝑦 = 𝐵
3		axrep6g 5296	. 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 1535   = wceq 1537   ∈ wcel 2106  ∃*wmo 2536  {cab 2712  ∃wrex 3068  Vcvv 3478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-rep 5285

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-v 3480

This theorem is referenced by:  abrexex  7986  iunexg  7987  qsexg  8814  wdomd  9619  cardiun  10020  rankcf  10815  sigaclci  34113  satf0suclem  35360  hbtlem1  43112  hbtlem7  43114  setpreimafvex  47308  fundcmpsurinj  47334  fundcmpsurbijinj  47335