MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brres Structured version   Visualization version   GIF version

Theorem brres 5572
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 5571 . 2 (𝐶𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
2 df-br 4810 . 2 (𝐵(𝑅𝐴)𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴))
3 df-br 4810 . . 3 (𝐵𝑅𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝑅)
43anbi2i 616 . 2 ((𝐵𝐴𝐵𝑅𝐶) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅))
51, 2, 43bitr4g 305 1 (𝐶𝑉 → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wcel 2155  cop 4340   class class class wbr 4809  cres 5279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-xp 5283  df-res 5289
This theorem is referenced by:  brresi  5574  dfima2  5650  axhcompl-zf  28246  fv1stcnv  32055  fv2ndcnv  32056  brcnvepres  34398  brres2  34399  eldmres  34400  elecres  34406  brinxprnres  34423  exanres  34427  eqres  34469  alrmomorn  34484  alrmomodm  34485  brxrn  34497  rnxrnres  34518  1cossres  34545  eldm1cossres  34571  brssrres  34615  dfdfat2  41808
  Copyright terms: Public domain W3C validator