Theorem brelrng 4893

Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)

Assertion

Ref	Expression
brelrng	|- ( ( A e. F /\ B e. G /\ A C B ) -> B e. ran C )

Proof of Theorem brelrng

Step	Hyp	Ref	Expression
1		brcnvg 4843	. . . . 5 |- ( ( B e. G /\ A e. F ) -> ( B `' C A <-> A C B ) )
2	1	ancoms 268	. . . 4 |- ( ( A e. F /\ B e. G ) -> ( B `' C A <-> A C B ) )
3	2	biimp3ar 1357	. . 3 |- ( ( A e. F /\ B e. G /\ A C B ) -> B `' C A )
4		breldmg 4868	. . . 4 |- ( ( B e. G /\ A e. F /\ B `' C A ) -> B e. dom `' C )
5	4	3com12 1209	. . 3 |- ( ( A e. F /\ B e. G /\ B `' C A ) -> B e. dom `' C )
6	3, 5	syld3an3 1294	. 2 |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. dom `' C )
7		df-rn 4670	. 2 |- ran C = dom `' C
8	6, 7	eleqtrrdi 2287	1 |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. ran C )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 980   e. wcel 2164   class class class wbr 4029   `'ccnv 4658   dom cdm 4659   ran crn 4660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-cnv 4667  df-dm 4669  df-rn 4670

This theorem is referenced by:  opelrng  4894  brelrn  4895  relelrn  4898  fvssunirng  5569  shftfvalg  10962  ovshftex  10963  shftfval  10965