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Theorem brres 4561
Description: Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
Hypothesis
Ref Expression
opelres.1 B V
Assertion
Ref Expression
brres (A(𝐶𝐷)B ↔ (A𝐶B A 𝐷))

Proof of Theorem brres
StepHypRef Expression
1 opelres.1 . . 3 B V
21opelres 4560 . 2 (⟨A, B (𝐶𝐷) ↔ (⟨A, B 𝐶 A 𝐷))
3 df-br 3756 . 2 (A(𝐶𝐷)B ↔ ⟨A, B (𝐶𝐷))
4 df-br 3756 . . 3 (A𝐶B ↔ ⟨A, B 𝐶)
54anbi1i 431 . 2 ((A𝐶B A 𝐷) ↔ (⟨A, B 𝐶 A 𝐷))
62, 3, 53bitr4i 201 1 (A(𝐶𝐷)B ↔ (A𝐶B A 𝐷))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   wcel 1390  Vcvv 2551  cop 3370   class class class wbr 3755  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-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-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-br 3756  df-opab 3810  df-xp 4294  df-res 4300
This theorem is referenced by:  dfres2  4601  dfima2  4613  poirr2  4660  cores  4767  resco  4768  rnco  4770  fnres  4958  fvres  5141  nfunsn  5150  1stconst  5784  2ndconst  5785
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