Theorem abrexexg 7973

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 5242, axrep6 5239, ax-rep 5231. See also abrexex2g 7976. There are partial converses under additional conditions, see for instance abnexg 7770. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2177, ax-11 2193, ax-12 2212, ax-pr 5390, ax-un 7751 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 3664	. . 3 ⊢ ∃*𝑦 𝑦 = 𝐵
2	1	ax-gen 1827	. 2 ⊢ ∀𝑥∃*𝑦 𝑦 = 𝐵
3		axrep6g 5242	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦 𝑦 = 𝐵) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
4	2, 3	mpan2 704	1 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∃*wmo 2562  {cab 2738  ∃wrex 3086  Vcvv 3450

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-rep 5231

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-v 3452

This theorem is used by:  abrexex  7974  iunexg  7975  qsexg  8795  wdomd  9578  cardiun  10066  rankcf  10865  sigaclci  34810  satf0suclem  36226  hbtlem1  44218  hbtlem7  44220  setpreimafvex  48558  fundcmpsurinj  48584  fundcmpsurbijinj  48585