Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmopab Structured version   GIF version

Theorem dmopab 4489
 Description: The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
Assertion
Ref Expression
dmopab dom {⟨x, y⟩ ∣ φ} = {xyφ}
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem dmopab
StepHypRef Expression
1 nfopab1 3817 . . 3 x{⟨x, y⟩ ∣ φ}
2 nfopab2 3818 . . 3 y{⟨x, y⟩ ∣ φ}
31, 2dfdmf 4471 . 2 dom {⟨x, y⟩ ∣ φ} = {xy x{⟨x, y⟩ ∣ φ}y}
4 df-br 3756 . . . . 5 (x{⟨x, y⟩ ∣ φ}y ↔ ⟨x, y {⟨x, y⟩ ∣ φ})
5 opabid 3985 . . . . 5 (⟨x, y {⟨x, y⟩ ∣ φ} ↔ φ)
64, 5bitri 173 . . . 4 (x{⟨x, y⟩ ∣ φ}yφ)
76exbii 1493 . . 3 (y x{⟨x, y⟩ ∣ φ}yyφ)
87abbii 2150 . 2 {xy x{⟨x, y⟩ ∣ φ}y} = {xyφ}
93, 8eqtri 2057 1 dom {⟨x, y⟩ ∣ φ} = {xyφ}
 Colors of variables: wff set class Syntax hints:   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  ⟨cop 3370   class class class wbr 3755  {copab 3808  dom cdm 4288 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-dm 4298 This theorem is referenced by:  dmopabss  4490  dmopab3  4491  fndmin  5217  dmoprab  5527
 Copyright terms: Public domain W3C validator