Theorem releldmi 5928

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 5924	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 690	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2108   class class class wbr 5119  dom cdm 5654  Rel wrel 5659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-dm 5664

This theorem is referenced by:  fpwwe2lem10  10654  fpwwe2lem11  10655  fpwwe2lem12  10656  rlimpm  15516  rlimdm  15567  iserex  15673  caucvgrlem2  15691  caucvgr  15692  caurcvg2  15694  caucvg  15695  fsumcvg3  15745  cvgcmpce  15834  climcnds  15867  trirecip  15879  ledm  18600  cmetcaulem  25240  ovoliunlem1  25455  mbflimlem  25620  dvaddf  25897  dvmulf  25898  dvcof  25904  dvcnv  25933  abelthlem5  26397  emcllem6  26963  lgamgulmlem4  26994  hlimcaui  31217  brfvrcld2  43716  sumnnodd  45659  climliminf  45835  stirlinglem12  46114  fouriersw  46260  rlimdmafv  47206  rlimdmafv2  47287