Theorem releldmi 5782

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 5778	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 689	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2111   class class class wbr 5030  dom cdm 5519  Rel wrel 5524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-dm 5529

This theorem is referenced by:  fpwwe2lem11  10051  fpwwe2lem12  10052  fpwwe2lem13  10053  rlimpm  14849  rlimdm  14900  iserex  15005  caucvgrlem2  15023  caucvgr  15024  caurcvg2  15026  caucvg  15027  fsumcvg3  15078  cvgcmpce  15165  climcnds  15198  trirecip  15210  ledm  17826  cmetcaulem  23892  ovoliunlem1  24106  mbflimlem  24271  dvaddf  24545  dvmulf  24546  dvcof  24551  dvcnv  24580  abelthlem5  25030  emcllem6  25586  lgamgulmlem4  25617  hlimcaui  29019  brfvrcld2  40393  sumnnodd  42272  climliminf  42448  stirlinglem12  42727  fouriersw  42873  rlimdmafv  43733  rlimdmafv2  43814