Theorem brelrng 4954

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 4902	. . . . 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 1380	. . 3 |- ( ( A e. F /\ B e. G /\ A C B ) -> B `' C A )
4		breldmg 4928	. . . 4 |- ( ( B e. G /\ A e. F /\ B `' C A ) -> B e. dom `' C )
5	4	3com12 1231	. . 3 |- ( ( A e. F /\ B e. G /\ B `' C A ) -> B e. dom `' C )
6	3, 5	syld3an3 1316	. 2 |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. dom `' C )
7		df-rn 4729	. 2 |- ran C = dom `' C
8	6, 7	eleqtrrdi 2323	1 |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. ran C )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   e. wcel 2200   class class class wbr 4082   `'ccnv 4717   dom cdm 4718   ran crn 4719

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-cnv 4726  df-dm 4728  df-rn 4729

This theorem is referenced by:  opelrng  4955  brelrn  4956  relelrn  4959  fvssunirng  5641  shftfvalg  11324  ovshftex  11325  shftfval  11327