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

Theorem unrab 4315
Description: Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
unrab ({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}

Proof of Theorem unrab
StepHypRef Expression
1 df-rab 3437 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 df-rab 3437 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
31, 2uneq12i 4166 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) = ({𝑥 ∣ (𝑥𝐴𝜑)} ∪ {𝑥 ∣ (𝑥𝐴𝜓)})
4 df-rab 3437 . . 3 {𝑥𝐴 ∣ (𝜑𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
5 unab 4308 . . . 4 ({𝑥 ∣ (𝑥𝐴𝜑)} ∪ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐴𝜓))}
6 andi 1010 . . . . 5 ((𝑥𝐴 ∧ (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐴𝜓)))
76abbii 2809 . . . 4 {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))} = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐴𝜓))}
85, 7eqtr4i 2768 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ∪ {𝑥 ∣ (𝑥𝐴𝜓)}) = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
94, 8eqtr4i 2768 . 2 {𝑥𝐴 ∣ (𝜑𝜓)} = ({𝑥 ∣ (𝑥𝐴𝜑)} ∪ {𝑥 ∣ (𝑥𝐴𝜓)})
103, 9eqtr4i 2768 1 ({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 395  wo 848   = wceq 1540  wcel 2108  {cab 2714  {crab 3436  cun 3949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-un 3956
This theorem is referenced by:  rabxm  4390  kmlem3  10193  hashbclem  14491  phiprmpw  16813  mndpsuppss  18778  efgsfo  19757  dsmmacl  21761  rrxmvallem  25438  mumul  27224  ppiub  27248  lgsquadlem2  27425  lrold  27935  edglnl  29160  numclwwlk3lem2lem  30402  3unrab  32522  zarclsun  33869  hasheuni  34086  measvuni  34215  aean  34245  subfacp1lem6  35190  lineunray  36148  cnambfre  37675  itg2addnclem2  37679  iblabsnclem  37690  orrabdioph  42792  sqrtcvallem1  43644  undisjrab  44325  dfsclnbgr6  47844
  Copyright terms: Public domain W3C validator