Theorem releldmi 5892

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 5888	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 690	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2113   class class class wbr 5093  dom cdm 5619  Rel wrel 5624

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-dm 5629

This theorem is referenced by:  fpwwe2lem10  10538  fpwwe2lem11  10539  fpwwe2lem12  10540  rlimpm  15409  rlimdm  15460  iserex  15566  caucvgrlem2  15584  caucvgr  15585  caurcvg2  15587  caucvg  15588  fsumcvg3  15638  cvgcmpce  15727  climcnds  15760  trirecip  15772  ledm  18498  cmetcaulem  25216  ovoliunlem1  25431  mbflimlem  25596  dvaddf  25873  dvmulf  25874  dvcof  25880  dvcnv  25909  abelthlem5  26373  emcllem6  26939  lgamgulmlem4  26970  hlimcaui  31218  brfvrcld2  43809  sumnnodd  45754  climliminf  45928  stirlinglem12  46207  fouriersw  46353  rlimdmafv  47301  rlimdmafv2  47382