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

Theorem rab2ex 5341
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 5340 . 2 𝐵 ∈ V
43rabex 5338 1 {𝑥𝐵𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  {crab 3435  Vcvv 3479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-in 3957  df-ss 3967  df-pw 4601
This theorem is referenced by:  gsumbagdiag  21952  psrlidm  21983  psrridm  21984  psrass1  21985  mdegmullem  26118  vtxdginducedm1lem4  29561  vtxdginducedm1  29562
  Copyright terms: Public domain W3C validator