Theorem relelrn 4847

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 4651	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
2		brrelex2 4652	. 2 |- ( ( Rel R /\ A R B ) -> B e. _V )
3		simpr 109	. 2 |- ( ( Rel R /\ A R B ) -> A R B )
4		brelrng 4842	. 2 |- ( ( A e. _V /\ B e. _V /\ A R B ) -> B e. ran R )
5	1, 2, 3, 4	syl3anc 1233	1 |- ( ( Rel R /\ A R B ) -> B e. ran R )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   _Vcvv 2730   class class class wbr 3989   ran crn 4612   Rel wrel 4616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622

This theorem is referenced by:  relelrnb  4849  relelrni  4851  relfvssunirn  5512