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Theorem brelrng 5947
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 5886 . . . . 5 ((𝐵𝐺𝐴𝐹) → (𝐵𝐶𝐴𝐴𝐶𝐵))
21ancoms 457 . . . 4 ((𝐴𝐹𝐵𝐺) → (𝐵𝐶𝐴𝐴𝐶𝐵))
32biimp3ar 1466 . . 3 ((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵𝐶𝐴)
4 breldmg 5916 . . . 4 ((𝐵𝐺𝐴𝐹𝐵𝐶𝐴) → 𝐵 ∈ dom 𝐶)
543com12 1120 . . 3 ((𝐴𝐹𝐵𝐺𝐵𝐶𝐴) → 𝐵 ∈ dom 𝐶)
63, 5syld3an3 1406 . 2 ((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ dom 𝐶)
7 df-rn 5693 . 2 ran 𝐶 = dom 𝐶
86, 7eleqtrrdi 2840 1 ((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084  wcel 2098   class class class wbr 5152  ccnv 5681  dom cdm 5682  ran crn 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-cnv 5690  df-dm 5692  df-rn 5693
This theorem is referenced by:  brelrn  5948  relelrn  5951  sossfld  6195  fvrn0  6932  pgpfaclem1  20045  perpln2  28535
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