Theorem abrexex 7998

Description: Existence of a class abstraction of existentially restricted sets. See the comment of abrexexg 7997. See also abrexex2 8005. (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 7997	. 2 ⊢ (𝐴 ∈ V → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
3	1, 2	ax-mp 5	1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-rep 5235

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-rex 3091  df-v 3456

This theorem is used by:  ab2rexex  8015  kmlem10  10265  cshwsexa  15003  shftfval  15252  dvdsrval  20621  cmpsublem  23747  cmpsub  23748  ptrescn  23988  addsproplem2  28386  negsid  28457  onaddscl  28693  recut  28910  elreno2  28911  satfvsuclem1  36291  fmlasuc0  36316  nmulprop  37117  heibor1lem  38935  pointsetN  40990  eldiophb  43961