Theorem relelrni 4838

Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)

Hypothesis

Ref	Expression
releldm.1	|- Rel R

Assertion

Ref	Expression
relelrni	|- ( A R B -> B e. ran R )

Proof of Theorem relelrni

Step	Hyp	Ref	Expression
1		releldm.1	. 2 |- Rel R
2		relelrn 4834	. 2 |- ( ( Rel R /\ A R B ) -> B e. ran R )
3	1, 2	mpan 421	1 |- ( A R B -> B e. ran R )

Syntax hints:   -> wi 4   e. wcel 2135   class class class wbr 3976   ran crn 4599   Rel wrel 4603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-xp 4604  df-rel 4605  df-cnv 4606  df-dm 4608  df-rn 4609

This theorem is referenced by: (None)