Theorem releldmi 5936

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 5932	. 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 2145   class class class wbr 5107  dom cdm 5659  Rel wrel 5664

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-dm 5669

This theorem is used by:  fpwwe2lem10  10653  fpwwe2lem11  10654  fpwwe2lem12  10655  rlimpm  15591  rlimdm  15642  iserex  15748  caucvgrlem2  15766  caucvgr  15767  caurcvg2  15769  caucvg  15770  fsumcvg3  15819  cvgcmpce  15909  climcnds  15944  trirecip  15956  ledm  18684  cmetcaulem  25522  ovoliunlem1  25736  mbflimlem  25901  dvaddf  26176  dvmulf  26177  dvcof  26182  dvcnv  26211  abelthlem5  26678  emcllem6  27245  lgamgulmlem4  27276  hlimcaui  31725  brfvrcld2  44540  sumnnodd  46468  climliminf  46642  stirlinglem12  46921  fouriersw  47067  rlimdmafv  48073  rlimdmafv2  48154