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Theorem relelrni 5597
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 5593 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
31, 2mpan 683 1 (𝐴𝑅𝐵𝐵 ∈ ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2166   class class class wbr 4874  ran crn 5344  Rel wrel 5348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-xp 5349  df-rel 5350  df-cnv 5351  df-dm 5353  df-rn 5354
This theorem is referenced by:  fpwwe2lem12  9779  lern  17579  brres2  34587  brfvrcld2  38826
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