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Theorem brelrng 4770
Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)
Assertion
Ref Expression
brelrng ((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)

Proof of Theorem brelrng
StepHypRef Expression
1 brcnvg 4720 . . . . 5 ((𝐵𝐺𝐴𝐹) → (𝐵𝐶𝐴𝐴𝐶𝐵))
21ancoms 266 . . . 4 ((𝐴𝐹𝐵𝐺) → (𝐵𝐶𝐴𝐴𝐶𝐵))
32biimp3ar 1324 . . 3 ((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵𝐶𝐴)
4 breldmg 4745 . . . 4 ((𝐵𝐺𝐴𝐹𝐵𝐶𝐴) → 𝐵 ∈ dom 𝐶)
543com12 1185 . . 3 ((𝐴𝐹𝐵𝐺𝐵𝐶𝐴) → 𝐵 ∈ dom 𝐶)
63, 5syld3an3 1261 . 2 ((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ dom 𝐶)
7 df-rn 4550 . 2 ran 𝐶 = dom 𝐶
86, 7eleqtrrdi 2233 1 ((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  w3a 962  wcel 1480   class class class wbr 3929  ccnv 4538  dom cdm 4539  ran crn 4540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547  df-dm 4549  df-rn 4550
This theorem is referenced by:  opelrng  4771  brelrn  4772  relelrn  4775  fvssunirng  5436  shftfvalg  10602  ovshftex  10603  shftfval  10605
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