Theorem releldmi 5930

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 5926	. 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 2145   class class class wbr 5103  dom cdm 5651  Rel wrel 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-dm 5661

This theorem is used by:  fpwwe2lem10  10706  fpwwe2lem11  10707  fpwwe2lem12  10708  rlimpm  15647  rlimdm  15698  iserex  15804  caucvgrlem2  15822  caucvgr  15823  caurcvg2  15825  caucvg  15826  fsumcvg3  15875  cvgcmpce  15965  climcnds  16000  trirecip  16012  ledm  18744  cmetcaulem  25589  ovoliunlem1  25803  mbflimlem  25968  dvaddf  26242  dvmulf  26243  dvcof  26248  dvcnv  26277  abelthlem5  26744  emcllem6  27310  lgamgulmlem4  27341  hlimcaui  31820  brfvrcld2  44651  sumnnodd  46586  climliminf  46760  stirlinglem12  47039  fouriersw  47185  rlimdmafv  48191  rlimdmafv2  48272