Theorem abrexex 7931

Description: Existence of a class abstraction of existentially restricted sets. See the comment of abrexexg 7929. See also abrexex2 7938. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
abrexex.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
abrexex	⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem abrexex

Step	Hyp	Ref	Expression
1		abrexex.1	. 2 ⊢ 𝐴 ∈ V
2		abrexexg 7929	. 2 ⊢ (𝐴 ∈ V → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
3	1, 2	ax-mp 5	1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V

Syntax hints:   = wceq 1541   ∈ wcel 2106  {cab 2708  ∃wrex 3069  Vcvv 3473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-rep 5278

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-rex 3070  df-v 3475

This theorem is referenced by:  ab2rexex  7948  kmlem10  10136  cshwsexa  14756  shftfval  14999  dvdsrval  20127  cmpsublem  22832  cmpsub  22833  ptrescn  23072  addsproplem2  27370  negsid  27431  satfvsuclem1  34179  fmlasuc0  34204  heibor1lem  36480  pointsetN  38415  eldiophb  41264