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Theorem brelrng 5951
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 5889 . . . . 5 ((𝐵𝐺𝐴𝐹) → (𝐵𝐶𝐴𝐴𝐶𝐵))
21ancoms 458 . . . 4 ((𝐴𝐹𝐵𝐺) → (𝐵𝐶𝐴𝐴𝐶𝐵))
32biimp3ar 1471 . . 3 ((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵𝐶𝐴)
4 breldmg 5919 . . . 4 ((𝐵𝐺𝐴𝐹𝐵𝐶𝐴) → 𝐵 ∈ dom 𝐶)
543com12 1123 . . 3 ((𝐴𝐹𝐵𝐺𝐵𝐶𝐴) → 𝐵 ∈ dom 𝐶)
63, 5syld3an3 1410 . 2 ((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ dom 𝐶)
7 df-rn 5695 . 2 ran 𝐶 = dom 𝐶
86, 7eleqtrrdi 2851 1 ((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086  wcel 2107   class class class wbr 5142  ccnv 5683  dom cdm 5684  ran crn 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-cnv 5692  df-dm 5694  df-rn 5695
This theorem is referenced by:  brelrn  5952  relelrn  5955  sossfld  6205  fvrn0  6935  pgpfaclem1  20102  perpln2  28720
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