Theorem opelrng 4861

Description: Membership of second member of an ordered pair in a range. (Contributed by Jim Kingdon, 26-Jan-2019.)

Assertion

Ref	Expression
opelrng	|- ( ( A e. F /\ B e. G /\ <. A , B >. e. C ) -> B e. ran C )

Proof of Theorem opelrng

Step	Hyp	Ref	Expression
1		df-br 4006	. 2 |- ( A C B <-> <. A , B >. e. C )
2		brelrng 4860	. 2 |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. ran C )
3	1, 2	syl3an3br 1279	1 |- ( ( A e. F /\ B e. G /\ <. A , B >. e. C ) -> B e. ran C )

Syntax hints:   -> wi 4   /\ w3a 978   e. wcel 2148   <.cop 3597   class class class wbr 4005   ran crn 4629

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-cnv 4636  df-dm 4638  df-rn 4639

This theorem is referenced by:  2ndrn  6186