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Theorem relelrn 4875
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 brrelex 4678 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
2 brrelex2 4679 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
3 simpr 110 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4 brelrng 4870 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
51, 2, 3, 4syl3anc 1248 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2158  Vcvv 2749   class class class wbr 4015  ran crn 4639  Rel wrel 4643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-dm 4648  df-rn 4649
This theorem is referenced by:  relelrnb  4877  relelrni  4879  relfvssunirn  5543
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