MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brinxp2 Structured version   Visualization version   GIF version

Theorem brinxp2 5387
Description: Intersection with cross product binary relation. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.) Group conjuncts and avoid df-3an 1102. (Revised by Peter Mazsa, 18-Sep-2022.)
Assertion
Ref Expression
brinxp2 (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶𝐴𝐷𝐵) ∧ 𝐶𝑅𝐷))

Proof of Theorem brinxp2
StepHypRef Expression
1 brin 4903 . 2 (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ (𝐶𝑅𝐷𝐶(𝐴 × 𝐵)𝐷))
2 ancom 450 . 2 ((𝐶𝑅𝐷𝐶(𝐴 × 𝐵)𝐷) ↔ (𝐶(𝐴 × 𝐵)𝐷𝐶𝑅𝐷))
3 brxp 5354 . . 3 (𝐶(𝐴 × 𝐵)𝐷 ↔ (𝐶𝐴𝐷𝐵))
43anbi1i 612 . 2 ((𝐶(𝐴 × 𝐵)𝐷𝐶𝑅𝐷) ↔ ((𝐶𝐴𝐷𝐵) ∧ 𝐶𝑅𝐷))
51, 2, 43bitri 288 1 (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶𝐴𝐷𝐵) ∧ 𝐶𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wcel 2157  cin 3775   class class class wbr 4851   × cxp 5316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-br 4852  df-opab 4914  df-xp 5324
This theorem is referenced by:  brinxp  5389  opelinxp  5390  fncnv  6176  erinxp  8059  fpwwe2lem8  9747  fpwwe2lem9  9748  fpwwe2lem12  9751  nqerf  10040  nqerid  10043  isstruct  16084  pwsle  16360  psss  17422  psssdm2  17423  pi1cpbl  23060  pi1grplem  23065  brres2  34354  inxpss  34400  inxpss3  34402  idinxpssinxp2  34406  inxp2  34444  inxpxrn  34468
  Copyright terms: Public domain W3C validator