Theorem releldmi 5897

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 5893	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 691	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2114   class class class wbr 5086  dom cdm 5624  Rel wrel 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-dm 5634

This theorem is referenced by:  fpwwe2lem10  10554  fpwwe2lem11  10555  fpwwe2lem12  10556  rlimpm  15453  rlimdm  15504  iserex  15610  caucvgrlem2  15628  caucvgr  15629  caurcvg2  15631  caucvg  15632  fsumcvg3  15682  cvgcmpce  15772  climcnds  15807  trirecip  15819  ledm  18547  cmetcaulem  25265  ovoliunlem1  25479  mbflimlem  25644  dvaddf  25919  dvmulf  25920  dvcof  25925  dvcnv  25954  abelthlem5  26413  emcllem6  26978  lgamgulmlem4  27009  hlimcaui  31322  brfvrcld2  44137  sumnnodd  46078  climliminf  46252  stirlinglem12  46531  fouriersw  46677  rlimdmafv  47637  rlimdmafv2  47718