Theorem abrexex 7916

Description: Existence of a class abstraction of existentially restricted sets. See the comment of abrexexg 7915. See also abrexex2 7923. (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 7915	. 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 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062  Vcvv 3442

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-rep 5226

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-rex 3063  df-v 3444

This theorem is referenced by:  ab2rexex  7933  kmlem10  10082  cshwsexa  14759  shftfval  15005  dvdsrval  20309  cmpsublem  23355  cmpsub  23356  ptrescn  23595  addsproplem2  27978  negsid  28049  onaddscl  28285  recut  28502  elreno2  28503  satfvsuclem1  35572  fmlasuc0  35597  heibor1lem  38054  pointsetN  40111  eldiophb  43108