Theorem releldmi 5946

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 5942	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 686	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2104   class class class wbr 5147  dom cdm 5675  Rel wrel 5680

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-dm 5685

This theorem is referenced by:  fpwwe2lem10  10637  fpwwe2lem11  10638  fpwwe2lem12  10639  rlimpm  15448  rlimdm  15499  iserex  15607  caucvgrlem2  15625  caucvgr  15626  caurcvg2  15628  caucvg  15629  fsumcvg3  15679  cvgcmpce  15768  climcnds  15801  trirecip  15813  ledm  18547  cmetcaulem  25036  ovoliunlem1  25251  mbflimlem  25416  dvaddf  25693  dvmulf  25694  dvcof  25700  dvcnv  25729  abelthlem5  26183  emcllem6  26741  lgamgulmlem4  26772  hlimcaui  30756  brfvrcld2  42745  sumnnodd  44644  climliminf  44820  stirlinglem12  45099  fouriersw  45245  rlimdmafv  46183  rlimdmafv2  46264