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Theorem relelrn 5886
Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
Assertion
Ref Expression
relelrn ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)

Proof of Theorem relelrn
StepHypRef Expression
1 brrelex1 5671 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
2 brrelex2 5672 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
3 simpr 485 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4 brelrng 5882 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
51, 2, 3, 4syl3anc 1370 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  Vcvv 3441   class class class wbr 5092  ran crn 5621  Rel wrel 5625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-xp 5626  df-rel 5627  df-cnv 5628  df-dm 5630  df-rn 5631
This theorem is referenced by:  relelrnb  5888  relelrni  5890  dirge  18418  metideq  32141  ntrneinex  42016
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