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Theorem brelrng 4592
 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 4543 . . . . 5 ((𝐵𝐺𝐴𝐹) → (𝐵𝐶𝐴𝐴𝐶𝐵))
21ancoms 259 . . . 4 ((𝐴𝐹𝐵𝐺) → (𝐵𝐶𝐴𝐴𝐶𝐵))
32biimp3ar 1252 . . 3 ((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵𝐶𝐴)
4 breldmg 4568 . . . 4 ((𝐵𝐺𝐴𝐹𝐵𝐶𝐴) → 𝐵 ∈ dom 𝐶)
543com12 1119 . . 3 ((𝐴𝐹𝐵𝐺𝐵𝐶𝐴) → 𝐵 ∈ dom 𝐶)
63, 5syld3an3 1191 . 2 ((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ dom 𝐶)
7 df-rn 4383 . 2 ran 𝐶 = dom 𝐶
86, 7syl6eleqr 2147 1 ((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102   ∧ w3a 896   ∈ wcel 1409   class class class wbr 3791  ◡ccnv 4371  dom cdm 4372  ran crn 4373 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-cnv 4380  df-dm 4382  df-rn 4383 This theorem is referenced by:  opelrng  4593  brelrn  4594  relelrn  4597  fvssunirng  5217  shftfvalg  9646  ovshftex  9647  shftfval  9649
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