Theorem releldmi 5821

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 5817	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 688	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2113   class class class wbr 5069  dom cdm 5558  Rel wrel 5563

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-dm 5568

This theorem is referenced by:  fpwwe2lem11  10065  fpwwe2lem12  10066  fpwwe2lem13  10067  rlimpm  14860  rlimdm  14911  iserex  15016  caucvgrlem2  15034  caucvgr  15035  caurcvg2  15037  caucvg  15038  fsumcvg3  15089  cvgcmpce  15176  climcnds  15209  trirecip  15221  ledm  17837  cmetcaulem  23894  ovoliunlem1  24106  mbflimlem  24271  dvaddf  24542  dvmulf  24543  dvcof  24548  dvcnv  24577  abelthlem5  25026  emcllem6  25581  lgamgulmlem4  25612  hlimcaui  29016  brfvrcld2  40043  sumnnodd  41917  climliminf  42093  stirlinglem12  42377  fouriersw  42523  rlimdmafv  43383  rlimdmafv2  43464