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Theorem brres 5986
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 5985 . 2 (𝐶𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
2 df-br 5143 . 2 (𝐵(𝑅𝐴)𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴))
3 df-br 5143 . . 3 (𝐵𝑅𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝑅)
43anbi2i 622 . 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 205  wa 395  wcel 2099  cop 4630   class class class wbr 5142  cres 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-res 5684
This theorem is referenced by:  brresi  5988  dfima2  6059  predres  6339  ttrclselem2  9741  axhcompl-zf  30795  fv1stcnv  35308  fv2ndcnv  35309  bj-idreseq  36577  bj-idreseqb  36578  brcnvepres  37674  brres2  37675  eldmres  37677  elrnres  37678  elecres  37684  brinxprnres  37699  exanres  37703  eqres  37748  alrmomorn  37766  alrmomodm  37767  brxrn  37783  rnxrnres  37808  1cossres  37838  brressn  37850  eldm1cossres  37869  brssrres  37913  disjres  38153  antisymrelres  38172  dfdfat2  46431
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