Theorem abrexex 7944

Description: Existence of a class abstraction of existentially restricted sets. See the comment of abrexexg 7942. See also abrexex2 7951. (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 7942	. 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 2708  ∃wrex 3054  Vcvv 3450

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 2702  ax-rep 5237

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-rex 3055  df-v 3452

This theorem is referenced by:  ab2rexex  7961  kmlem10  10120  cshwsexa  14796  shftfval  15043  dvdsrval  20277  cmpsublem  23293  cmpsub  23294  ptrescn  23533  addsproplem2  27884  negsid  27954  onaddscl  28181  recut  28354  0reno  28355  satfvsuclem1  35353  fmlasuc0  35378  heibor1lem  37810  pointsetN  39742  eldiophb  42752