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Theorem releldmi 5900
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 5896 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
31, 2mpan 689 1 (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107   class class class wbr 5104  dom cdm 5631  Rel wrel 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5637  df-rel 5638  df-dm 5641
This theorem is referenced by:  fpwwe2lem10  10510  fpwwe2lem11  10511  fpwwe2lem12  10512  rlimpm  15317  rlimdm  15368  iserex  15476  caucvgrlem2  15494  caucvgr  15495  caurcvg2  15497  caucvg  15498  fsumcvg3  15549  cvgcmpce  15638  climcnds  15671  trirecip  15683  ledm  18414  cmetcaulem  24575  ovoliunlem1  24789  mbflimlem  24954  dvaddf  25229  dvmulf  25230  dvcof  25235  dvcnv  25264  abelthlem5  25717  emcllem6  26273  lgamgulmlem4  26304  hlimcaui  29977  brfvrcld2  41727  sumnnodd  43626  climliminf  43802  stirlinglem12  44081  fouriersw  44227  rlimdmafv  45164  rlimdmafv2  45245
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