Theorem releldmi 5929

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 5925	. 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 2144   class class class wbr 5102  dom cdm 5650  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-dm 5660

This theorem is used by:  fpwwe2lem10  10728  fpwwe2lem11  10729  fpwwe2lem12  10730  rlimpm  15672  rlimdm  15723  iserex  15829  caucvgrlem2  15847  caucvgr  15848  caurcvg2  15850  caucvg  15851  fsumcvg3  15900  cvgcmpce  15990  climcnds  16025  trirecip  16037  ledm  18769  cmetcaulem  25614  ovoliunlem1  25828  mbflimlem  25993  dvaddf  26267  dvmulf  26268  dvcof  26273  dvcnv  26302  abelthlem5  26767  emcllem6  27333  lgamgulmlem4  27364  hlimcaui  31843  brfvrcld2  44785  ishfstruct  46121  sumnnodd  46735  climliminf  46909  stirlinglem12  47188  fouriersw  47334  rlimdmafv  48340  rlimdmafv2  48421