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Theorem relelrni 5904
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 5900 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
31, 2mpan 688 1 (𝐴𝑅𝐵𝐵 ∈ ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106   class class class wbr 5105  ran crn 5634  Rel wrel 5638
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-xp 5639  df-rel 5640  df-cnv 5641  df-dm 5643  df-rn 5644
This theorem is referenced by:  fpwwe2lem11  10576  lern  18479  brres2  36718  brfvrcld2  41945
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