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

Theorem releldmi 5322
Description: The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)
Hypothesis
Ref Expression
releldm.1 Rel 𝑅
Assertion
Ref Expression
releldmi (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)

Proof of Theorem releldmi
StepHypRef Expression
1 releldm.1 . 2 Rel 𝑅
2 releldm 5318 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
31, 2mpan 705 1 (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987   class class class wbr 4613  dom cdm 5074  Rel wrel 5079
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  df-rel 5081  df-dm 5084
This theorem is referenced by:  fpwwe2lem11  9406  fpwwe2lem12  9407  fpwwe2lem13  9408  rlimpm  14165  rlimdm  14216  iserex  14321  caucvgrlem2  14339  caucvgr  14340  caurcvg2  14342  caucvg  14343  fsumcvg3  14393  cvgcmpce  14477  climcnds  14508  trirecip  14520  ledm  17145  cmetcaulem  22994  ovoliunlem1  23177  mbflimlem  23340  dvaddf  23611  dvmulf  23612  dvcof  23617  dvcnv  23644  abelthlem5  24093  emcllem6  24627  lgamgulmlem4  24658  hlimcaui  27942  brfvrcld2  37465  sumnnodd  39266  stirlinglem12  39609  fouriersw  39755  rlimdmafv  40561
  Copyright terms: Public domain W3C validator