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 690	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2113   class class class wbr 5098  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 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-dm 5634

This theorem is referenced by:  fpwwe2lem10  10551  fpwwe2lem11  10552  fpwwe2lem12  10553  rlimpm  15423  rlimdm  15474  iserex  15580  caucvgrlem2  15598  caucvgr  15599  caurcvg2  15601  caucvg  15602  fsumcvg3  15652  cvgcmpce  15741  climcnds  15774  trirecip  15786  ledm  18513  cmetcaulem  25244  ovoliunlem1  25459  mbflimlem  25624  dvaddf  25901  dvmulf  25902  dvcof  25908  dvcnv  25937  abelthlem5  26401  emcllem6  26967  lgamgulmlem4  26998  hlimcaui  31311  brfvrcld2  43929  sumnnodd  45872  climliminf  46046  stirlinglem12  46325  fouriersw  46471  rlimdmafv  47419  rlimdmafv2  47500