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Theorem relelrni 5942
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 5938 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
31, 2mpan 687 1 (𝐴𝑅𝐵𝐵 ∈ ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098   class class class wbr 5141  ran crn 5670  Rel wrel 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680
This theorem is referenced by:  fpwwe2lem11  10638  lern  18556  brres2  37649  brfvrcld2  43016
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