Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  relelrn Structured version   GIF version

Theorem relelrn 4513
 Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
Assertion
Ref Expression
relelrn ((Rel 𝑅 A𝑅B) → B ran 𝑅)

Proof of Theorem relelrn
StepHypRef Expression
1 brrelex 4325 . 2 ((Rel 𝑅 A𝑅B) → A V)
2 brrelex2 4326 . 2 ((Rel 𝑅 A𝑅B) → B V)
3 simpr 103 . 2 ((Rel 𝑅 A𝑅B) → A𝑅B)
4 brelrng 4508 . 2 ((A V B V A𝑅B) → B ran 𝑅)
51, 2, 3, 4syl3anc 1134 1 ((Rel 𝑅 A𝑅B) → B ran 𝑅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  Vcvv 2551   class class class wbr 3755  ran crn 4289  Rel wrel 4293 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299 This theorem is referenced by:  relelrnb  4515  relelrni  4517  relfvssunirn  5134
 Copyright terms: Public domain W3C validator