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Theorem relelrn 4570
 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 4382 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
2 brrelex2 4383 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
3 simpr 103 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4 brelrng 4565 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
51, 2, 3, 4syl3anc 1135 1 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1393  Vcvv 2557   class class class wbr 3764  ran crn 4346  Rel wrel 4350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356 This theorem is referenced by:  relelrnb  4572  relelrni  4574  relfvssunirn  5191
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