Theorem rabexf 45378

Description: Separation Scheme in terms of a restricted class abstraction. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
rabexf.1	⊢ Ⅎ𝑥𝐴
rabexf.2	⊢ 𝐴 ∈ 𝑉

Assertion

Ref	Expression
rabexf	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V

Proof of Theorem rabexf

Step	Hyp	Ref	Expression
1		rabexf.2	. 2 ⊢ 𝐴 ∈ 𝑉
2		rabexf.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	2	rabexgf 45269	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
4	1, 3	ax-mp 5	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V

Syntax hints:   ∈ wcel 2113  Ⅎwnfc 2883  {crab 3399  Vcvv 3440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3400  df-v 3442  df-in 3908  df-ss 3918

This theorem is referenced by:  limsupequzmpt2  45962  liminfequzmpt2  46035  fsupdm  47086  finfdm  47090