Theorem brelrng 5238

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 5189	. . . . 5 |- ( ( B e. G /\ A e. F ) -> ( B `' C A <-> A C B ) )
2	1	ancoms 453	. . . 4 |- ( ( A e. F /\ B e. G ) -> ( B `' C A <-> A C B ) )
3	2	biimp3ar 1329	. . 3 |- ( ( A e. F /\ B e. G /\ A C B ) -> B `' C A )
4		breldmg 5214	. . . 4 |- ( ( B e. G /\ A e. F /\ B `' C A ) -> B e. dom `' C )
5	4	3com12 1200	. . 3 |- ( ( A e. F /\ B e. G /\ B `' C A ) -> B e. dom `' C )
6	3, 5	syld3an3 1273	. 2 |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. dom `' C )
7		df-rn 5016	. 2 |- ran C = dom `' C
8	6, 7	syl6eleqr 2566	1 |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. ran C )

Syntax hints:   -> wi 4   <-> wb 184   /\ w3a 973   e. wcel 1767   class class class wbr 4453   `'ccnv 5004   dom cdm 5005   ran crn 5006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-cnv 5013  df-dm 5015  df-rn 5016

This theorem is referenced by:  brelrn  5239  relelrn  5242  sossfld  5460  fvrn0  5894  pgpfaclem1  17004  perpln2  23936