Theorem brelrng 4988

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 4936	. . . . 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 1383	. . 3 |- ( ( A e. F /\ B e. G /\ A C B ) -> B `' C A )
4		breldmg 4962	. . . 4 |- ( ( B e. G /\ A e. F /\ B `' C A ) -> B e. dom `' C )
5	4	3com12 1234	. . 3 |- ( ( A e. F /\ B e. G /\ B `' C A ) -> B e. dom `' C )
6	3, 5	syld3an3 1319	. 2 |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. dom `' C )
7		df-rn 4760	. 2 |- ran C = dom `' C
8	6, 7	eleqtrrdi 2326	1 |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. ran C )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1005   e. wcel 2203   class class class wbr 4109   `'ccnv 4748   dom cdm 4749   ran crn 4750

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-cnv 4757  df-dm 4759  df-rn 4760

This theorem is referenced by:  opelrng  4989  brelrn  4990  relelrn  4993  fvssunirng  5685  shftfvalg  11503  ovshftex  11504  shftfval  11506