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Theorem brel 5128
Description: Two things in a binary relation belong to the relation's domain. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
brel.1 𝑅 ⊆ (𝐶 × 𝐷)
Assertion
Ref Expression
brel (𝐴𝑅𝐵 → (𝐴𝐶𝐵𝐷))

Proof of Theorem brel
StepHypRef Expression
1 brel.1 . . 3 𝑅 ⊆ (𝐶 × 𝐷)
21ssbri 4657 . 2 (𝐴𝑅𝐵𝐴(𝐶 × 𝐷)𝐵)
3 brxp 5107 . 2 (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐷))
42, 3sylib 208 1 (𝐴𝑅𝐵 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wss 3555   class class class wbr 4613   × cxp 5072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080
This theorem is referenced by:  brab2a  5129  brab2ga  5155  soirri  5481  sotri  5482  sotri2  5484  sotri3  5485  ndmovord  6777  ndmovordi  6778  swoer  7717  brecop2  7786  ecopovsym  7794  ecopovtrn  7795  hartogslem1  8391  nlt1pi  9672  indpi  9673  nqerf  9696  ordpipq  9708  lterpq  9736  ltexnq  9741  ltbtwnnq  9744  ltrnq  9745  prnmadd  9763  genpcd  9772  nqpr  9780  1idpr  9795  ltexprlem4  9805  ltexpri  9809  ltaprlem  9810  prlem936  9813  reclem2pr  9814  reclem3pr  9815  reclem4pr  9816  suplem1pr  9818  suplem2pr  9819  supexpr  9820  recexsrlem  9868  addgt0sr  9869  mulgt0sr  9870  mappsrpr  9873  map2psrpr  9875  supsrlem  9876  supsr  9877  ltresr  9905  dfle2  11924  dflt2  11925  dvdszrcl  14912  letsr  17148  hmphtop  21491  vcex  27282  brtxp2  31630  brpprod3a  31635
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