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Theorem relelrn 5943
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 5728 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
2 brrelex2 5729 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
3 simpr 483 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4 brelrng 5939 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
51, 2, 3, 4syl3anc 1369 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2104  Vcvv 3472   class class class wbr 5147  ran crn 5676  Rel wrel 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686
This theorem is referenced by:  relelrnb  5945  relelrni  5947  dirge  18560  metideq  33171  ntrneinex  43130
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