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Theorem releldmi 5795
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 5791 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
31, 2mpan 689 1 (𝐴𝑅𝐵𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114   class class class wbr 5042  dom cdm 5532  Rel wrel 5537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-dm 5542
This theorem is referenced by:  fpwwe2lem11  10051  fpwwe2lem12  10052  fpwwe2lem13  10053  rlimpm  14848  rlimdm  14899  iserex  15004  caucvgrlem2  15022  caucvgr  15023  caurcvg2  15025  caucvg  15026  fsumcvg3  15077  cvgcmpce  15164  climcnds  15197  trirecip  15209  ledm  17825  cmetcaulem  23890  ovoliunlem1  24104  mbflimlem  24269  dvaddf  24543  dvmulf  24544  dvcof  24549  dvcnv  24578  abelthlem5  25028  emcllem6  25584  lgamgulmlem4  25615  hlimcaui  29017  brfvrcld2  40323  sumnnodd  42211  climliminf  42387  stirlinglem12  42666  fouriersw  42812  rlimdmafv  43672  rlimdmafv2  43753
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