Theorem releldmi 5846

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 5842	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 686	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2108   class class class wbr 5070  dom cdm 5580  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-dm 5590

This theorem is referenced by:  fpwwe2lem10  10327  fpwwe2lem11  10328  fpwwe2lem12  10329  rlimpm  15137  rlimdm  15188  iserex  15296  caucvgrlem2  15314  caucvgr  15315  caurcvg2  15317  caucvg  15318  fsumcvg3  15369  cvgcmpce  15458  climcnds  15491  trirecip  15503  ledm  18223  cmetcaulem  24357  ovoliunlem1  24571  mbflimlem  24736  dvaddf  25011  dvmulf  25012  dvcof  25017  dvcnv  25046  abelthlem5  25499  emcllem6  26055  lgamgulmlem4  26086  hlimcaui  29499  brfvrcld2  41189  sumnnodd  43061  climliminf  43237  stirlinglem12  43516  fouriersw  43662  rlimdmafv  44556  rlimdmafv2  44637