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Theorem brres 4751
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 ↔ (𝐴𝐶𝐵𝐴𝐷))

Proof of Theorem brres
StepHypRef Expression
1 opelres.1 . . 3 𝐵 ∈ V
21opelres 4750 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))
3 df-br 3868 . 2 (𝐴(𝐶𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷))
4 df-br 3868 . . 3 (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
54anbi1i 447 . 2 ((𝐴𝐶𝐵𝐴𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))
62, 3, 53bitr4i 211 1 (𝐴(𝐶𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐴𝐷))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wcel 1445  Vcvv 2633  cop 3469   class class class wbr 3867  cres 4469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-xp 4473  df-res 4479
This theorem is referenced by:  dfres2  4797  dfima2  4809  poirr2  4857  cores  4968  resco  4969  rnco  4971  fnres  5164  fvres  5364  nfunsn  5373  1stconst  6024  2ndconst  6025
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