Theorem releldmi 5856

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 5852	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 687	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2110   class class class wbr 5079  dom cdm 5590  Rel wrel 5595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-xp 5596  df-rel 5597  df-dm 5600

This theorem is referenced by:  fpwwe2lem10  10397  fpwwe2lem11  10398  fpwwe2lem12  10399  rlimpm  15207  rlimdm  15258  iserex  15366  caucvgrlem2  15384  caucvgr  15385  caurcvg2  15387  caucvg  15388  fsumcvg3  15439  cvgcmpce  15528  climcnds  15561  trirecip  15573  ledm  18306  cmetcaulem  24450  ovoliunlem1  24664  mbflimlem  24829  dvaddf  25104  dvmulf  25105  dvcof  25110  dvcnv  25139  abelthlem5  25592  emcllem6  26148  lgamgulmlem4  26179  hlimcaui  29594  brfvrcld2  41270  sumnnodd  43142  climliminf  43318  stirlinglem12  43597  fouriersw  43743  rlimdmafv  44637  rlimdmafv2  44718