Theorem releldmi 5947

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 5943	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 687	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2105   class class class wbr 5148  dom cdm 5676  Rel wrel 5681

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-dm 5686

This theorem is referenced by:  fpwwe2lem10  10638  fpwwe2lem11  10639  fpwwe2lem12  10640  rlimpm  15449  rlimdm  15500  iserex  15608  caucvgrlem2  15626  caucvgr  15627  caurcvg2  15629  caucvg  15630  fsumcvg3  15680  cvgcmpce  15769  climcnds  15802  trirecip  15814  ledm  18548  cmetcaulem  25037  ovoliunlem1  25252  mbflimlem  25417  dvaddf  25694  dvmulf  25695  dvcof  25701  dvcnv  25730  abelthlem5  26184  emcllem6  26742  lgamgulmlem4  26773  hlimcaui  30757  brfvrcld2  42746  sumnnodd  44645  climliminf  44821  stirlinglem12  45100  fouriersw  45246  rlimdmafv  46184  rlimdmafv2  46265