MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rab2ex Structured version   Visualization version   GIF version

Theorem rab2ex 4949
Description: A class abstraction based on a class abstraction based on a set is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
Hypotheses
Ref Expression
rab2ex.1 𝐵 = {𝑦𝐴𝜓}
rab2ex.2 𝐴 ∈ V
Assertion
Ref Expression
rab2ex {𝑥𝐵𝜑} ∈ V
Distinct variable groups:   𝑥,𝐵   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem rab2ex
StepHypRef Expression
1 rab2ex.1 . . 3 𝐵 = {𝑦𝐴𝜓}
2 rab2ex.2 . . 3 𝐴 ∈ V
31, 2rabex2 4948 . 2 𝐵 ∈ V
43rabex 4946 1 {𝑥𝐵𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-in 3730  df-ss 3737
This theorem is referenced by:  gsumbagdiag  19591  psrlidm  19618  psrridm  19619  psrass1  19620  mdegmullem  24058  vtxdginducedm1  26674
  Copyright terms: Public domain W3C validator