Theorem relelrn 4993

Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)

Assertion

Ref	Expression
relelrn	|- ( ( Rel R /\ A R B ) -> B e. ran R )

Proof of Theorem relelrn

Step	Hyp	Ref	Expression
1		brrelex 4790	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
2		brrelex2 4791	. 2 |- ( ( Rel R /\ A R B ) -> B e. _V )
3		simpr 110	. 2 |- ( ( Rel R /\ A R B ) -> A R B )
4		brelrng 4988	. 2 |- ( ( A e. _V /\ B e. _V /\ A R B ) -> B e. ran R )
5	1, 2, 3, 4	syl3anc 1274	1 |- ( ( Rel R /\ A R B ) -> B e. ran R )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2203   _Vcvv 2813   class class class wbr 4109   ran crn 4750   Rel wrel 4754

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-ral 2525  df-rex 2526  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-xp 4755  df-rel 4756  df-cnv 4757  df-dm 4759  df-rn 4760

This theorem is referenced by:  relelrnb  4995  relelrni  4997  relfvssunirn  5686