Theorem opelrn 4781

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 3938	. 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 4780	. 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 1481   _Vcvv 2689   <.cop 3535   class class class wbr 3937   ran crn 4548

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-cnv 4555  df-dm 4557  df-rn 4558

This theorem is referenced by: (None)