Theorem rabexf 45581

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 45472	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
4	1, 3	ax-mp 5	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V

Syntax hints:   ∈ wcel 2119  Ⅎwnfc 2886  {crab 3391  Vcvv 3431

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-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-rab 3392  df-v 3433  df-in 3890  df-ss 3900

This theorem is referenced by:  limsupequzmpt2  46161  liminfequzmpt2  46234  fsupdm  47285  finfdm  47289