Theorem releldmi 5283

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 5279	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 702	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 1977   class class class wbr 4583  dom cdm 5038  Rel wrel 5043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048

This theorem is referenced by:  fpwwe2lem11  9341  fpwwe2lem12  9342  fpwwe2lem13  9343  rlimpm  14079  rlimdm  14130  iserex  14235  caucvgrlem2  14253  caucvgr  14254  caurcvg2  14256  caucvg  14257  fsumcvg3  14307  cvgcmpce  14391  climcnds  14422  trirecip  14434  ledm  17047  cmetcaulem  22894  ovoliunlem1  23077  mbflimlem  23240  dvaddf  23511  dvmulf  23512  dvcof  23517  dvcnv  23544  abelthlem5  23993  emcllem6  24527  lgamgulmlem4  24558  hlimcaui  27477  brfvrcld2  37003  sumnnodd  38697  stirlinglem12  38978  fouriersw  39124  rlimdmafv  39906