Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  brelrn Structured version   Visualization version   GIF version

Theorem brelrn 5388
 Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)
Hypotheses
Ref Expression
brelrn.1 𝐴 ∈ V
brelrn.2 𝐵 ∈ V
Assertion
Ref Expression
brelrn (𝐴𝐶𝐵𝐵 ∈ ran 𝐶)

Proof of Theorem brelrn
StepHypRef Expression
1 brelrn.1 . 2 𝐴 ∈ V
2 brelrn.2 . 2 𝐵 ∈ V
3 brelrng 5387 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
41, 2, 3mp3an12 1454 1 (𝐴𝐶𝐵𝐵 ∈ ran 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2030  Vcvv 3231   class class class wbr 4685  ran crn 5144 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-cnv 5151  df-dm 5153  df-rn 5154 This theorem is referenced by:  opelrn  5389  dfco2a  5673  cores  5676  dffun9  5955  funcnv  5996  rntpos  7410  aceq3lem  8981  axdclem  9379  axdclem2  9380  cotr2g  13761  shftfval  13854  psdmrn  17254  metustexhalf  22408  itg1addlem4  23511
 Copyright terms: Public domain W3C validator