Theorem releldmi 5937

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 5933	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 703	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   class class class wbr 5106  dom cdm 5655  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-dm 5665

This theorem is used by:  fpwwe2lem10  10752  fpwwe2lem11  10753  fpwwe2lem12  10754  rlimpm  15696  rlimdm  15747  iserex  15853  caucvgrlem2  15871  caucvgr  15872  caurcvg2  15874  caucvg  15875  fsumcvg3  15924  cvgcmpce  16014  climcnds  16049  trirecip  16061  ledm  18793  cmetcaulem  25639  ovoliunlem1  25853  mbflimlem  26018  dvaddf  26292  dvmulf  26293  dvcof  26298  dvcnv  26327  abelthlem5  26792  emcllem6  27358  lgamgulmlem4  27389  hlimcaui  31868  brfvrcld2  44884  ishfstruct  46226  sumnnodd  46847  climliminf  47021  stirlinglem12  47300  fouriersw  47446  rlimdmafv  48452  rlimdmafv2  48533