Theorem abrexex 7911

Description: Existence of a class abstraction of existentially restricted sets. See the comment of abrexexg 7910. See also abrexex2 7918. (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 7910	. 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 1547   ∈ wcel 2119  {cab 2718  ∃wrex 3064  Vcvv 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-rep 5206

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2719  df-cleq 2732  df-clel 2815  df-rex 3065  df-v 3434

This theorem is referenced by:  ab2rexex  7928  kmlem10  10080  cshwsexa  14784  shftfval  15030  dvdsrval  20339  cmpsublem  23389  cmpsub  23390  ptrescn  23629  addsproplem2  27987  negsid  28058  onaddscl  28294  recut  28511  elreno2  28512  satfvsuclem1  35594  fmlasuc0  35619  heibor1lem  38183  pointsetN  40240  eldiophb  43213