Theorem abrexex 7976

Description: Existence of a class abstraction of existentially restricted sets. See the comment of abrexexg 7974. See also abrexex2 7983. (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 7974	. 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 1534   ∈ wcel 2099  {cab 2703  ∃wrex 3060  Vcvv 3462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-rep 5290

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-mo 2529  df-clab 2704  df-cleq 2718  df-clel 2803  df-rex 3061  df-v 3464

This theorem is referenced by:  ab2rexex  7993  kmlem10  10202  cshwsexa  14832  shftfval  15075  dvdsrval  20343  cmpsublem  23394  cmpsub  23395  ptrescn  23634  addsproplem2  27984  negsid  28050  recut  28347  0reno  28348  satfvsuclem1  35187  fmlasuc0  35212  heibor1lem  37510  pointsetN  39440  eldiophb  42414