Theorem relelrn 4902

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 4703	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
2		brrelex2 4704	. 2 |- ( ( Rel R /\ A R B ) -> B e. _V )
3		simpr 110	. 2 |- ( ( Rel R /\ A R B ) -> A R B )
4		brelrng 4897	. 2 |- ( ( A e. _V /\ B e. _V /\ A R B ) -> B e. ran R )
5	1, 2, 3, 4	syl3anc 1249	1 |- ( ( Rel R /\ A R B ) -> B e. ran R )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   _Vcvv 2763   class class class wbr 4033   ran crn 4664   Rel wrel 4668

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-dm 4673  df-rn 4674

This theorem is referenced by:  relelrnb  4904  relelrni  4906  relfvssunirn  5574