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

Theorem opres 4564
 Description: Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypothesis
Ref Expression
opres.1 B V
Assertion
Ref Expression
opres (A 𝐷 → (⟨A, B (𝐶𝐷) ↔ ⟨A, B 𝐶))

Proof of Theorem opres
StepHypRef Expression
1 opres.1 . . 3 B V
21opelres 4560 . 2 (⟨A, B (𝐶𝐷) ↔ (⟨A, B 𝐶 A 𝐷))
32rbaib 829 1 (A 𝐷 → (⟨A, B (𝐶𝐷) ↔ ⟨A, B 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370   ↾ cres 4290 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-bndl 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-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  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-opab 3810  df-xp 4294  df-res 4300 This theorem is referenced by:  resieq  4565  2elresin  4953
 Copyright terms: Public domain W3C validator