Theorem releldmi 5888

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	⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Proof of Theorem releldmi

Step	Hyp	Ref	Expression
1		releldm.1	. 2 ⊢ Rel 𝑅
2		releldm 5884	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 690	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2111   class class class wbr 5091  dom cdm 5616  Rel wrel 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-dm 5626

This theorem is referenced by:  fpwwe2lem10  10528  fpwwe2lem11  10529  fpwwe2lem12  10530  rlimpm  15404  rlimdm  15455  iserex  15561  caucvgrlem2  15579  caucvgr  15580  caurcvg2  15582  caucvg  15583  fsumcvg3  15633  cvgcmpce  15722  climcnds  15755  trirecip  15767  ledm  18493  cmetcaulem  25213  ovoliunlem1  25428  mbflimlem  25593  dvaddf  25870  dvmulf  25871  dvcof  25877  dvcnv  25906  abelthlem5  26370  emcllem6  26936  lgamgulmlem4  26967  hlimcaui  31211  brfvrcld2  43724  sumnnodd  45669  climliminf  45843  stirlinglem12  46122  fouriersw  46268  rlimdmafv  47207  rlimdmafv2  47288