Theorem opelrn 4838

Description: Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.)

Hypotheses

Ref	Expression
brelrn.1	|- A e. _V
brelrn.2	|- B e. _V

Assertion

Ref	Expression
opelrn	|- ( <. A , B >. e. C -> B e. ran C )

Proof of Theorem opelrn

Step	Hyp	Ref	Expression
1		df-br 3983	. 2 |- ( A C B <-> <. A , B >. e. C )
2		brelrn.1	. . 3 |- A e. _V
3		brelrn.2	. . 3 |- B e. _V
4	2, 3	brelrn 4837	. 2 |- ( A C B -> B e. ran C )
5	1, 4	sylbir 134	1 |- ( <. A , B >. e. C -> B e. ran C )

Syntax hints:   -> wi 4   e. wcel 2136   _Vcvv 2726   <.cop 3579   class class class wbr 3982   ran crn 4605

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by: (None)