Theorem releldmi 5857

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 5853	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 687	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2106   class class class wbr 5074  dom cdm 5589  Rel wrel 5594

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-dm 5599

This theorem is referenced by:  fpwwe2lem10  10396  fpwwe2lem11  10397  fpwwe2lem12  10398  rlimpm  15209  rlimdm  15260  iserex  15368  caucvgrlem2  15386  caucvgr  15387  caurcvg2  15389  caucvg  15390  fsumcvg3  15441  cvgcmpce  15530  climcnds  15563  trirecip  15575  ledm  18308  cmetcaulem  24452  ovoliunlem1  24666  mbflimlem  24831  dvaddf  25106  dvmulf  25107  dvcof  25112  dvcnv  25141  abelthlem5  25594  emcllem6  26150  lgamgulmlem4  26181  hlimcaui  29598  brfvrcld2  41300  sumnnodd  43171  climliminf  43347  stirlinglem12  43626  fouriersw  43772  rlimdmafv  44669  rlimdmafv2  44750