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Theorem relelrni 5787
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 (𝐴𝑅𝐵𝐵 ∈ ran 𝑅)

Proof of Theorem relelrni
StepHypRef Expression
1 releldm.1 . 2 Rel 𝑅
2 relelrn 5783 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
31, 2mpan 689 1 (𝐴𝑅𝐵𝐵 ∈ ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112   class class class wbr 5033  ran crn 5524  Rel wrel 5528
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534
This theorem is referenced by:  fpwwe2lem12  10056  lern  17830  brres2  35682  brfvrcld2  40380
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