Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  brelrng Structured version   Unicode version

Theorem brelrng 5026
 Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)
Assertion
Ref Expression
brelrng

Proof of Theorem brelrng
StepHypRef Expression
1 brcnvg 4977 . . . . 5
21ancoms 454 . . . 4
32biimp3ar 1365 . . 3
4 breldmg 5002 . . . 4
543com12 1209 . . 3
63, 5syld3an3 1309 . 2
7 df-rn 4807 . 2
86, 7syl6eleqr 2517 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   w3a 982   wcel 1872   class class class wbr 4366  ccnv 4795   cdm 4796   crn 4797 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-opab 4426  df-cnv 4804  df-dm 4806  df-rn 4807 This theorem is referenced by:  brelrn  5027  relelrn  5030  sossfld  5245  fvrn0  5847  pgpfaclem1  17657  perpln2  24698
 Copyright terms: Public domain W3C validator