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 702	1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2145   class class class wbr 5105  dom cdm 5652  Rel wrel 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-dm 5662

This theorem is referenced by:  fpwwe2lem10  10613  fpwwe2lem11  10614  fpwwe2lem12  10615  rlimpm  15541  rlimdm  15592  iserex  15698  caucvgrlem2  15716  caucvgr  15717  caurcvg2  15719  caucvg  15720  fsumcvg3  15770  cvgcmpce  15860  climcnds  15895  trirecip  15907  ledm  18636  cmetcaulem  25408  ovoliunlem1  25622  mbflimlem  25787  dvaddf  26062  dvmulf  26063  dvcof  26068  dvcnv  26097  abelthlem5  26556  emcllem6  27123  lgamgulmlem4  27154  hlimcaui  31497  brfvrcld2  44280  sumnnodd  46204  climliminf  46378  stirlinglem12  46657  fouriersw  46803  rlimdmafv  47769  rlimdmafv2  47850