Theorem releldmi 5915

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 5911	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 690	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2109   class class class wbr 5110  dom cdm 5641  Rel wrel 5646

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-dm 5651

This theorem is referenced by:  fpwwe2lem10  10600  fpwwe2lem11  10601  fpwwe2lem12  10602  rlimpm  15473  rlimdm  15524  iserex  15630  caucvgrlem2  15648  caucvgr  15649  caurcvg2  15651  caucvg  15652  fsumcvg3  15702  cvgcmpce  15791  climcnds  15824  trirecip  15836  ledm  18556  cmetcaulem  25195  ovoliunlem1  25410  mbflimlem  25575  dvaddf  25852  dvmulf  25853  dvcof  25859  dvcnv  25888  abelthlem5  26352  emcllem6  26918  lgamgulmlem4  26949  hlimcaui  31172  brfvrcld2  43688  sumnnodd  45635  climliminf  45811  stirlinglem12  46090  fouriersw  46236  rlimdmafv  47182  rlimdmafv2  47263