Theorem releldmi 5903

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 5899	. 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 5085  dom cdm 5631  Rel wrel 5636

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 2708  ax-sep 5231  ax-pr 5375

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-dm 5641

This theorem is referenced by:  fpwwe2lem10  10563  fpwwe2lem11  10564  fpwwe2lem12  10565  rlimpm  15462  rlimdm  15513  iserex  15619  caucvgrlem2  15637  caucvgr  15638  caurcvg2  15640  caucvg  15641  fsumcvg3  15691  cvgcmpce  15781  climcnds  15816  trirecip  15828  ledm  18556  cmetcaulem  25255  ovoliunlem1  25469  mbflimlem  25634  dvaddf  25909  dvmulf  25910  dvcof  25915  dvcnv  25944  abelthlem5  26400  emcllem6  26964  lgamgulmlem4  26995  hlimcaui  31307  brfvrcld2  44119  sumnnodd  46060  climliminf  46234  stirlinglem12  46513  fouriersw  46659  rlimdmafv  47625  rlimdmafv2  47706