Theorem releldmi 5905

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 5901	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
3	1, 2	mpan 691	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2114   class class class wbr 5100  dom cdm 5632  Rel wrel 5637

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-dm 5642

This theorem is referenced by:  fpwwe2lem10  10563  fpwwe2lem11  10564  fpwwe2lem12  10565  rlimpm  15435  rlimdm  15486  iserex  15592  caucvgrlem2  15610  caucvgr  15611  caurcvg2  15613  caucvg  15614  fsumcvg3  15664  cvgcmpce  15753  climcnds  15786  trirecip  15798  ledm  18525  cmetcaulem  25256  ovoliunlem1  25471  mbflimlem  25636  dvaddf  25913  dvmulf  25914  dvcof  25920  dvcnv  25949  abelthlem5  26413  emcllem6  26979  lgamgulmlem4  27010  hlimcaui  31323  brfvrcld2  44042  sumnnodd  45984  climliminf  46158  stirlinglem12  46437  fouriersw  46583  rlimdmafv  47531  rlimdmafv2  47612