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Theorem brres 6007
Description: Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.) Commute the consequent. (Revised by Peter Mazsa, 24-Sep-2022.)
Assertion
Ref Expression
brres (𝐶𝑉 → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))

Proof of Theorem brres
StepHypRef Expression
1 opelres 6006 . 2 (𝐶𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
2 df-br 5149 . 2 (𝐵(𝑅𝐴)𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴))
3 df-br 5149 . . 3 (𝐵𝑅𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝑅)
43anbi2i 623 . 2 ((𝐵𝐴𝐵𝑅𝐶) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅))
51, 2, 43bitr4g 314 1 (𝐶𝑉 → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  cop 4637   class class class wbr 5148  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-res 5701
This theorem is referenced by:  brresi  6009  dfima2  6082  predres  6362  ttrclselem2  9764  axhcompl-zf  31027  fv1stcnv  35758  fv2ndcnv  35759  bj-idreseq  37145  bj-idreseqb  37146  brcnvepres  38249  brres2  38250  eldmres  38252  elrnres  38253  elecres  38259  brinxprnres  38273  exanres  38277  eqres  38322  alrmomorn  38340  alrmomodm  38341  brxrn  38356  rnxrnres  38381  1cossres  38411  brressn  38423  eldm1cossres  38442  brssrres  38486  disjres  38726  antisymrelres  38745  dfdfat2  47078
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