Theorem brelrng 5180

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 5131	. . . . 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 1320	. . 3 |- ( ( A e. F /\ B e. G /\ A C B ) -> B `' C A )
4		breldmg 5156	. . . 4 |- ( ( B e. G /\ A e. F /\ B `' C A ) -> B e. dom `' C )
5	4	3com12 1192	. . 3 |- ( ( A e. F /\ B e. G /\ B `' C A ) -> B e. dom `' C )
6	3, 5	syld3an3 1264	. 2 |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. dom `' C )
7		df-rn 4962	. 2 |- ran C = dom `' C
8	6, 7	syl6eleqr 2553	1 |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. ran C )

Syntax hints:   -> wi 4   <-> wb 184   /\ w3a 965   e. wcel 1758   class class class wbr 4403   `'ccnv 4950   dom cdm 4951   ran crn 4952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-cnv 4959  df-dm 4961  df-rn 4962

This theorem is referenced by:  brelrn  5181  relelrn  5184  sossfld  5396  fvrn0  5824  pgpfaclem1  16714  perpln2  23255