Theorem releldmi 5813

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 5809	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 688	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2110   class class class wbr 5059  dom cdm 5550  Rel wrel 5555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-xp 5556  df-rel 5557  df-dm 5560

This theorem is referenced by:  fpwwe2lem11  10056  fpwwe2lem12  10057  fpwwe2lem13  10058  rlimpm  14851  rlimdm  14902  iserex  15007  caucvgrlem2  15025  caucvgr  15026  caurcvg2  15028  caucvg  15029  fsumcvg3  15080  cvgcmpce  15167  climcnds  15200  trirecip  15212  ledm  17828  cmetcaulem  23885  ovoliunlem1  24097  mbflimlem  24262  dvaddf  24533  dvmulf  24534  dvcof  24539  dvcnv  24568  abelthlem5  25017  emcllem6  25572  lgamgulmlem4  25603  hlimcaui  29007  brfvrcld2  40030  sumnnodd  41903  climliminf  42079  stirlinglem12  42363  fouriersw  42509  rlimdmafv  43369  rlimdmafv2  43450