Theorem releldmi 5362

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	|- ( A R B -> A e. dom R )

Proof of Theorem releldmi

Step	Hyp	Ref	Expression
1		releldm.1	. 2 |- Rel R
2		releldm 5358	. 2 |- ( ( Rel R /\ A R B ) -> A e. dom R )
3	1, 2	mpan 706	1 |- ( A R B -> A e. dom R )

Syntax hints:   -> wi 4   e. wcel 1990   class class class wbr 4653   dom cdm 5114   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124

This theorem is referenced by:  fpwwe2lem11  9462  fpwwe2lem12  9463  fpwwe2lem13  9464  rlimpm  14231  rlimdm  14282  iserex  14387  caucvgrlem2  14405  caucvgr  14406  caurcvg2  14408  caucvg  14409  fsumcvg3  14460  cvgcmpce  14550  climcnds  14583  trirecip  14595  ledm  17224  cmetcaulem  23086  ovoliunlem1  23270  mbflimlem  23434  dvaddf  23705  dvmulf  23706  dvcof  23711  dvcnv  23740  abelthlem5  24189  emcllem6  24727  lgamgulmlem4  24758  hlimcaui  28093  brfvrcld2  37984  sumnnodd  39862  climliminf  40038  stirlinglem12  40302  fouriersw  40448  rlimdmafv  41257