Theorem abrexex 7941

Description: Existence of a class abstraction of existentially restricted sets. See the comment of abrexexg 7939. See also abrexex2 7948. (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 7939	. 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 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-rep 5234

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-rex 3054  df-v 3449

This theorem is referenced by:  ab2rexex  7958  kmlem10  10113  cshwsexa  14789  shftfval  15036  dvdsrval  20270  cmpsublem  23286  cmpsub  23287  ptrescn  23526  addsproplem2  27877  negsid  27947  onaddscl  28174  recut  28347  0reno  28348  satfvsuclem1  35346  fmlasuc0  35371  heibor1lem  37803  pointsetN  39735  eldiophb  42745