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Theorem brres 6003
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 6002 . 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 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 2107  cop 4631   class class class wbr 5142  cres 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-res 5696
This theorem is referenced by:  brresi  6005  dfima2  6079  predres  6359  ttrclselem2  9767  axhcompl-zf  31018  fv1stcnv  35778  fv2ndcnv  35779  bj-idreseq  37164  bj-idreseqb  37165  brcnvepres  38269  brres2  38270  eldmres  38272  elrnres  38273  elecres  38279  brinxprnres  38293  exanres  38297  eqres  38342  alrmomorn  38360  alrmomodm  38361  brxrn  38376  rnxrnres  38401  1cossres  38431  brressn  38443  eldm1cossres  38462  brssrres  38506  disjres  38746  antisymrelres  38765  dfdfat2  47145
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